Theorem fin1a2lem1 10311

Description: Lemma for fin1a2 10326. (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 7755	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		suceq 6383	. . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3		fin1a2lem.a	. . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4		suceq 6383	. . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
5	4	cbvmptv 5190	. . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
6	3, 5	eqtri 2760	. . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
7	2, 6	fvmptg 6937	. 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆‘𝐴) = suc 𝐴)
8	1, 7	mpdan 688	1 ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5167  Oncon0 6315  suc csuc 6317  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fv 6498

This theorem is referenced by:  fin1a2lem2  10312  fin1a2lem6  10316  onsucf1o  43715