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Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2eq12d | Structured version Visualization version GIF version |
Description: Equality deduction for function value, analogous to fveq12d 6670. (Contributed by AV, 4-Sep-2022.) |
Ref | Expression |
---|---|
afv2eq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
afv2eq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
afv2eq12d | ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | afv2eq12d.1 | . . . 4 ⊢ (𝜑 → 𝐹 = 𝐺) | |
2 | afv2eq12d.2 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 1, 2 | dfateq12d 43316 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | eqidd 2820 | . . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥) | |
5 | 2, 1, 4 | breq123d 5071 | . . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥)) |
6 | 5 | iotabidv 6332 | . . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥)) |
7 | 1 | rneqd 5801 | . . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺) |
8 | 7 | unieqd 4840 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺) |
9 | 8 | pweqd 4542 | . . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺) |
10 | 3, 6, 9 | ifbieq12d 4492 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)) |
11 | df-afv2 43399 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
12 | df-afv2 43399 | . 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺) | |
13 | 10, 11, 12 | 3eqtr4g 2879 | 1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ifcif 4465 𝒫 cpw 4537 ∪ cuni 4830 class class class wbr 5057 ran crn 5549 ℩cio 6305 defAt wdfat 43306 ''''cafv2 43398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-dfat 43309 df-afv2 43399 |
This theorem is referenced by: afv2eq1 43406 afv2eq2 43407 csbafv212g 43409 |
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