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Theorem fin1a2lem1 10294
Description: Lemma for fin1a2 10309. (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 onsuc 7738 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
2 suceq 6381 . . 3 (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3 fin1a2lem.a . . . 4 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4 suceq 6381 . . . . 5 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
54cbvmptv 5216 . . . 4 (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
63, 5eqtri 2764 . . 3 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
72, 6fvmptg 6943 . 2 ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆𝐴) = suc 𝐴)
81, 7mpdan 685 1 (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  cmpt 5186  Oncon0 6315  suc csuc 6317  cfv 6493
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6445  df-fun 6495  df-fv 6501
This theorem is referenced by:  fin1a2lem2  10295  fin1a2lem6  10299  onsucf1o  41510
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