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Theorem fin1a2lem1 9815
Description: Lemma for fin1a2 9830. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.a 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
Assertion
Ref Expression
fin1a2lem1 (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)

Proof of Theorem fin1a2lem1
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 suceloni 7512 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
2 suceq 6228 . . 3 (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3 fin1a2lem.a . . . 4 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4 suceq 6228 . . . . 5 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
54cbvmptv 5136 . . . 4 (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
63, 5eqtri 2824 . . 3 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
72, 6fvmptg 6747 . 2 ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆𝐴) = suc 𝐴)
81, 7mpdan 686 1 (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  cmpt 5113  Oncon0 6163  suc csuc 6165  cfv 6328
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-ord 6166  df-on 6167  df-suc 6169  df-iota 6287  df-fun 6330  df-fv 6336
This theorem is referenced by:  fin1a2lem2  9816  fin1a2lem6  9820
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