Theorem fin1a2lem1 10313

Description: Lemma for fin1a2 10328. (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 7753	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		suceq 6378	. . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3		fin1a2lem.a	. . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4		suceq 6378	. . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
5	4	cbvmptv 5176	. . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
6	3, 5	eqtri 2762	. . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
7	2, 6	fvmptg 6933	. 2 ⊢ ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆‘𝐴) = suc 𝐴)
8	1, 7	mpdan 693	1 ⊢ (𝐴 ∈ On → (𝑆‘𝐴) = suc 𝐴)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ↦ cmpt 5153  Oncon0 6310  suc csuc 6312  ‘cfv 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-ord 6313  df-on 6314  df-suc 6316  df-iota 6441  df-fun 6487  df-fv 6493

This theorem is referenced by:  fin1a2lem2  10314  fin1a2lem6  10318  onsucf1o  43717