Theorem fin1a2lem1 10440

Description: Lemma for fin1a2 10455. (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 7831	. 2 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
2		suceq 6450	. . 3 ⊢ (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3		fin1a2lem.a	. . . 4 ⊢ 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4		suceq 6450	. . . . 5 ⊢ (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
5	4	cbvmptv 5255	. . . 4 ⊢ (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
6	3, 5	eqtri 2765	. . 3 ⊢ 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
7	2, 6	fvmptg 7014	. 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 1540   ∈ wcel 2108   ↦ cmpt 5225  Oncon0 6384  suc csuc 6386  ‘cfv 6561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fv 6569

This theorem is referenced by:  fin1a2lem2  10441  fin1a2lem6  10445  onsucf1o  43285