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Theorem afv2eq12d 47227
Description: Equality deduction for function value, analogous to fveq12d 6913. (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 47138 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
4 eqidd 2738 . . . . 5 (𝜑𝑥 = 𝑥)
52, 1, 4breq123d 5157 . . . 4 (𝜑 → (𝐴𝐹𝑥𝐵𝐺𝑥))
65iotabidv 6545 . . 3 (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
71rneqd 5949 . . . . 5 (𝜑 → ran 𝐹 = ran 𝐺)
87unieqd 4920 . . . 4 (𝜑 ran 𝐹 = ran 𝐺)
98pweqd 4617 . . 3 (𝜑 → 𝒫 ran 𝐹 = 𝒫 ran 𝐺)
103, 6, 9ifbieq12d 4554 . 2 (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺))
11 df-afv2 47221 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
12 df-afv2 47221 . 2 (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺)
1310, 11, 123eqtr4g 2802 1 (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  ifcif 4525  𝒫 cpw 4600   cuni 4907   class class class wbr 5143  ran crn 5686  cio 6512   defAt wdfat 47128  ''''cafv2 47220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-dfat 47131  df-afv2 47221
This theorem is referenced by:  afv2eq1  47228  afv2eq2  47229  csbafv212g  47231
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