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Theorem afv2eq12d 42111
Description: Equality deduction for function value, analogous to fveq12d 6444. (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 42022 . . 3 (𝜑 → (𝐹 defAt 𝐴𝐺 defAt 𝐵))
4 eqidd 2826 . . . . 5 (𝜑𝑥 = 𝑥)
52, 1, 4breq123d 4889 . . . 4 (𝜑 → (𝐴𝐹𝑥𝐵𝐺𝑥))
65iotabidv 6111 . . 3 (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
71rneqd 5589 . . . . 5 (𝜑 → ran 𝐹 = ran 𝐺)
87unieqd 4670 . . . 4 (𝜑 ran 𝐹 = ran 𝐺)
98pweqd 4385 . . 3 (𝜑 → 𝒫 ran 𝐹 = 𝒫 ran 𝐺)
103, 6, 9ifbieq12d 4335 . 2 (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺))
11 df-afv2 42105 . 2 (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ran 𝐹)
12 df-afv2 42105 . 2 (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ran 𝐺)
1310, 11, 123eqtr4g 2886 1 (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1656  ifcif 4308  𝒫 cpw 4380   cuni 4660   class class class wbr 4875  ran crn 5347  cio 6088   defAt wdfat 42012  ''''cafv2 42104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-dfat 42015  df-afv2 42105
This theorem is referenced by:  afv2eq1  42112  afv2eq2  42113  csbafv212g  42115
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