Theorem Arroweqd ≪ | index | src | ≫

theorem Arroweqd (_G: wff) (_A1 _A2 _B1 _B2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> _B1 == _B2 $ >
  $ _G -> Arrow _A1 _B1 == Arrow _A2 _B2 $;


_G -> (isfun f <-> isfun f)
_G -> Dom f == Dom f
_G -> _A1 == _A2
_G -> (Dom f == _A1 <-> Dom f == _A2)
_G -> (isfun f /\ Dom f == _A1 <-> isfun f /\ Dom f == _A2)
_G -> Ran f == Ran f
_G -> _B1 == _B2
_G -> (Ran f C_ _B1 <-> Ran f C_ _B2)
_G -> (isfun f /\ Dom f == _A1 /\ Ran f C_ _B1 <-> isfun f /\ Dom f == _A2 /\ Ran f C_ _B2)
_G -> {f | isfun f /\ Dom f == _A1 /\ Ran f C_ _B1} == {f | isfun f /\ Dom f == _A2 /\ Ran f C_ _B2}
_G -> Arrow _A1 _B1 == Arrow _A2 _B2

Step	Hyp	Ref	Expression
1		biidd	_G -> (isfun f <-> isfun f)
2		eqsidd	_G -> Dom f == Dom f
3		hyp _Ah	_G -> _A1 == _A2
4	2, 3	eqseqd	_G -> (Dom f == _A1 <-> Dom f == _A2)
5	1, 4	aneqd	_G -> (isfun f /\ Dom f == _A1 <-> isfun f /\ Dom f == _A2)
6		eqsidd	_G -> Ran f == Ran f
7		hyp _Bh	_G -> _B1 == _B2
8	6, 7	sseqd	_G -> (Ran f C_ _B1 <-> Ran f C_ _B2)
9	5, 8	aneqd	_G -> (isfun f /\ Dom f == _A1 /\ Ran f C_ _B1 <-> isfun f /\ Dom f == _A2 /\ Ran f C_ _B2)
10	9	abeqd	_G -> {f \| isfun f /\ Dom f == _A1 /\ Ran f C_ _B1} == {f \| isfun f /\ Dom f == _A2 /\ Ran f C_ _B2}
11	10	conv Arrow	_G -> Arrow _A1 _B1 == Arrow _A2 _B2

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_set (elab, ax_8)