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Theorem fin1a2lem1 10437
Description: Lemma for fin1a2 10452. (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 7830 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
2 suceq 6451 . . 3 (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3 fin1a2lem.a . . . 4 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4 suceq 6451 . . . . 5 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
54cbvmptv 5260 . . . 4 (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
63, 5eqtri 2762 . . 3 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
72, 6fvmptg 7013 . 2 ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆𝐴) = suc 𝐴)
81, 7mpdan 687 1 (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cmpt 5230  Oncon0 6385  suc csuc 6387  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-ord 6388  df-on 6389  df-suc 6391  df-iota 6515  df-fun 6564  df-fv 6570
This theorem is referenced by:  fin1a2lem2  10438  fin1a2lem6  10442  onsucf1o  43261
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