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Theorem fin1a2lem1 9810
Description: Lemma for fin1a2 9825. (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 7517 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
2 suceq 6249 . . 3 (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3 fin1a2lem.a . . . 4 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4 suceq 6249 . . . . 5 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
54cbvmptv 5160 . . . 4 (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
63, 5eqtri 2841 . . 3 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
72, 6fvmptg 6759 . 2 ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆𝐴) = suc 𝐴)
81, 7mpdan 683 1 (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cmpt 5137  Oncon0 6184  suc csuc 6186  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  fin1a2lem2  9811  fin1a2lem6  9815
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