Mathbox for Alexander van der Vekens |
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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 43400 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | eqidd 2821 | . . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥) | |
5 | 2, 1, 4 | breq123d 5073 | . . . 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 4845 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺) |
9 | 8 | pweqd 4551 | . . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺) |
10 | 3, 6, 9 | ifbieq12d 4487 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)) |
11 | df-afv2 43483 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
12 | df-afv2 43483 | . 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺) | |
13 | 10, 11, 12 | 3eqtr4g 2880 | 1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ifcif 4460 𝒫 cpw 4532 ∪ cuni 4831 class class class wbr 5059 ran crn 5549 ℩cio 6305 defAt wdfat 43390 ''''cafv2 43482 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 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 43393 df-afv2 43483 |
This theorem is referenced by: afv2eq1 43490 afv2eq2 43491 csbafv212g 43493 |
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