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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 6899. (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 46569 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | eqidd 2726 | . . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥) | |
5 | 2, 1, 4 | breq123d 5157 | . . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥)) |
6 | 5 | iotabidv 6527 | . . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥)) |
7 | 1 | rneqd 5934 | . . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺) |
8 | 7 | unieqd 4916 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺) |
9 | 8 | pweqd 4615 | . . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺) |
10 | 3, 6, 9 | ifbieq12d 4552 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)) |
11 | df-afv2 46652 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
12 | df-afv2 46652 | . 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺) | |
13 | 10, 11, 12 | 3eqtr4g 2790 | 1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ifcif 4524 𝒫 cpw 4598 ∪ cuni 4903 class class class wbr 5143 ran crn 5673 ℩cio 6493 defAt wdfat 46559 ''''cafv2 46651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-iota 6495 df-fun 6545 df-dfat 46562 df-afv2 46652 |
This theorem is referenced by: afv2eq1 46659 afv2eq2 46660 csbafv212g 46662 |
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