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Theorem afv2eq12d 46495
Description: Equality deduction for function value, analogous to fveq12d 6892. (Contributed by AV, 4-Sep-2022.)
Hypotheses
Ref Expression
afv2eq12d.1 (𝜑𝐹 = 𝐺)
afv2eq12d.2 (𝜑𝐴 = 𝐵)
Assertion
Ref Expression
afv2eq12d (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Proof of Theorem afv2eq12d
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 afv2eq12d.1 . . . 4 (𝜑𝐹 = 𝐺)
2 afv2eq12d.2 . . . 4 (𝜑𝐴 = 𝐵)
31, 2dfateq12d 46406 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
4 eqidd 2727 . . . . 5 (𝜑𝑥 = 𝑥)
52, 1, 4breq123d 5155 . . . 4 (𝜑 → (𝐴𝐹𝑥𝐵𝐺𝑥))
65iotabidv 6521 . . 3 (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
71rneqd 5931 . . . . 5 (𝜑 → ran 𝐹 = ran 𝐺)
87unieqd 4915 . . . 4 (𝜑 ran 𝐹 = ran 𝐺)
98pweqd 4614 . . 3 (𝜑 → 𝒫 ran 𝐹 = 𝒫 ran 𝐺)
103, 6, 9ifbieq12d 4551 . 2 (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺))
11 df-afv2 46489 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
12 df-afv2 46489 . 2 (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺)
1310, 11, 123eqtr4g 2791 1 (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ifcif 4523  𝒫 cpw 4597   cuni 4902   class class class wbr 5141  ran crn 5670  cio 6487   defAt wdfat 46396  ''''cafv2 46488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-dfat 46399  df-afv2 46489
This theorem is referenced by:  afv2eq1  46496  afv2eq2  46497  csbafv212g  46499
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