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Theorem afv2eq12d 46658
Description: Equality deduction for function value, analogous to fveq12d 6899. (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 46569 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
4 eqidd 2726 . . . . 5 (𝜑𝑥 = 𝑥)
52, 1, 4breq123d 5157 . . . 4 (𝜑 → (𝐴𝐹𝑥𝐵𝐺𝑥))
65iotabidv 6527 . . 3 (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
71rneqd 5934 . . . . 5 (𝜑 → ran 𝐹 = ran 𝐺)
87unieqd 4916 . . . 4 (𝜑 ran 𝐹 = ran 𝐺)
98pweqd 4615 . . 3 (𝜑 → 𝒫 ran 𝐹 = 𝒫 ran 𝐺)
103, 6, 9ifbieq12d 4552 . 2 (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺))
11 df-afv2 46652 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
12 df-afv2 46652 . 2 (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺)
1310, 11, 123eqtr4g 2790 1 (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  ifcif 4524  𝒫 cpw 4598   cuni 4903   class class class wbr 5143  ran crn 5673  cio 6493   defAt wdfat 46559  ''''cafv2 46651
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 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-dfat 46562  df-afv2 46652
This theorem is referenced by:  afv2eq1  46659  afv2eq2  46660  csbafv212g  46662
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