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Theorem fin1a2lem1 9260
 Description: Lemma for fin1a2 9275. (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 7055 . 2 (𝐴 ∈ On → suc 𝐴 ∈ On)
2 suceq 5828 . . 3 (𝑎 = 𝐴 → suc 𝑎 = suc 𝐴)
3 fin1a2lem.a . . . 4 𝑆 = (𝑥 ∈ On ↦ suc 𝑥)
4 suceq 5828 . . . . 5 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
54cbvmptv 4783 . . . 4 (𝑥 ∈ On ↦ suc 𝑥) = (𝑎 ∈ On ↦ suc 𝑎)
63, 5eqtri 2673 . . 3 𝑆 = (𝑎 ∈ On ↦ suc 𝑎)
72, 6fvmptg 6319 . 2 ((𝐴 ∈ On ∧ suc 𝐴 ∈ On) → (𝑆𝐴) = suc 𝐴)
81, 7mpdan 703 1 (𝐴 ∈ On → (𝑆𝐴) = suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030   ↦ cmpt 4762  Oncon0 5761  suc csuc 5763  ‘cfv 5926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-ord 5764  df-on 5765  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934 This theorem is referenced by:  fin1a2lem2  9261  fin1a2lem6  9265
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