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Theorem afv2eq12d 43634
Description: Equality deduction for function value, analogous to fveq12d 6665. (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 43545 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
4 eqidd 2825 . . . . 5 (𝜑𝑥 = 𝑥)
52, 1, 4breq123d 5066 . . . 4 (𝜑 → (𝐴𝐹𝑥𝐵𝐺𝑥))
65iotabidv 6327 . . 3 (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
71rneqd 5795 . . . . 5 (𝜑 → ran 𝐹 = ran 𝐺)
87unieqd 4838 . . . 4 (𝜑 ran 𝐹 = ran 𝐺)
98pweqd 4540 . . 3 (𝜑 → 𝒫 ran 𝐹 = 𝒫 ran 𝐺)
103, 6, 9ifbieq12d 4476 . 2 (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺))
11 df-afv2 43628 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
12 df-afv2 43628 . 2 (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺)
1310, 11, 123eqtr4g 2884 1 (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  ifcif 4449  𝒫 cpw 4521   cuni 4824   class class class wbr 5052  ran crn 5543  cio 6300   defAt wdfat 43535  ''''cafv2 43627
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-iota 6302  df-fun 6345  df-dfat 43538  df-afv2 43628
This theorem is referenced by:  afv2eq1  43635  afv2eq2  43636  csbafv212g  43638
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