Theorem isfss ≪ | index | src | ≫ 

theorem isfss (A B: set): $ A C_ B -> isfun B -> isfun A $;
StepHypRefExpression
1 anllr 
A C_ B /\ isfun B /\ a, b e. A /\ a, c e. A -> isfun B
2 an3l 
A C_ B /\ isfun B /\ a, b e. A /\ a, c e. A -> A C_ B
3 anlr 
A C_ B /\ isfun B /\ a, b e. A /\ a, c e. A -> a, b e. A
4 2, 3 sseld 
A C_ B /\ isfun B /\ a, b e. A /\ a, c e. A -> a, b e. B
5 anr 
A C_ B /\ isfun B /\ a, b e. A /\ a, c e. A -> a, c e. A
6 2, 5 sseld 
A C_ B /\ isfun B /\ a, b e. A /\ a, c e. A -> a, c e. B
7 1, 4, 6 isfd 
A C_ B /\ isfun B /\ a, b e. A /\ a, c e. A -> b = c
8 7 exp 
A C_ B /\ isfun B /\ a, b e. A -> a, c e. A -> b = c
9 8 exp 
A C_ B /\ isfun B -> a, b e. A -> a, c e. A -> b = c
10 9 iald 
A C_ B /\ isfun B -> A. c (a, b e. A -> a, c e. A -> b = c)
11 10 iald 
A C_ B /\ isfun B -> A. b A. c (a, b e. A -> a, c e. A -> b = c)
12 11 iald 
A C_ B /\ isfun B -> A. a A. b A. c (a, b e. A -> a, c e. A -> b = c)
13 12 conv isfun 
A C_ B /\ isfun B -> isfun A
14 13 exp 
A C_ B -> isfun B -> isfun A

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), axs_the (theid, the0), axs_peano (peano2, addeq, muleq)