Theorem sreclem ≪ | index | src | ≫

theorem sreclem (f n: nat) {x: nat} (a: nat x):
  $ f = \. x e. upto n, a -> size (Dom f) = n $;

Step	Hyp	Ref	Expression
1		sizeupto	size (upto n) = n
2		dmrlam	Dom (\. x e. upto n, a) == upto n
3		nseq	f = \. x e. upto n, a -> f == \. x e. upto n, a
4	3	dmeqd	f = \. x e. upto n, a -> Dom f == Dom (\. x e. upto n, a)
5	2, 4	syl6eqs	f = \. x e. upto n, a -> Dom f == upto n
6	5	sizeeqd	f = \. x e. upto n, a -> size (Dom f) = size (upto n)
7	1, 6	syl6eq	f = \. x e. upto n, a -> size (Dom f) = n

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 (peano1, peano2, peano5, addeq, muleq, add0, addS, mul0, mulS)