Theorem fin1a2lem1 10302

Description: Lemma for fin1a2 10317. (Contributed by Stefan O'Rear, 7-Nov-2014.)

Hypothesis

Ref	Expression
fin1a2lem.a	⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)

Assertion

Ref	Expression
fin1a2lem1	⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴)

Proof of Theorem fin1a2lem1

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		onsuc 7752	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		suceq 6382	. . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3		fin1a2lem.a	. . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4		suceq 6382	. . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
5	4	cbvmptv 5199	. . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
6	3, 5	eqtri 2756	. . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
7	2, 6	fvmptg 6936	. 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆‘𝐴) = suc 𝐴)
8	1, 7	mpdan 687	1 ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113   ↦ cmpt 5176  Oncon0 6314  suc csuc 6316  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fv 6497

This theorem is referenced by:  fin1a2lem2  10303  fin1a2lem6  10307  onsucf1o  43429