![]() |
Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > afv2eq12d | Structured version Visualization version GIF version |
Description: Equality deduction for function value, analogous to fveq12d 6652. (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 43682 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | eqidd 2799 | . . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥) | |
5 | 2, 1, 4 | breq123d 5044 | . . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥)) |
6 | 5 | iotabidv 6308 | . . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥)) |
7 | 1 | rneqd 5772 | . . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺) |
8 | 7 | unieqd 4814 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺) |
9 | 8 | pweqd 4516 | . . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺) |
10 | 3, 6, 9 | ifbieq12d 4452 | . 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)) |
11 | df-afv2 43765 | . 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) | |
12 | df-afv2 43765 | . 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺) | |
13 | 10, 11, 12 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ifcif 4425 𝒫 cpw 4497 ∪ cuni 4800 class class class wbr 5030 ran crn 5520 ℩cio 6281 defAt wdfat 43672 ''''cafv2 43764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-dfat 43675 df-afv2 43765 |
This theorem is referenced by: afv2eq1 43772 afv2eq2 43773 csbafv212g 43775 |
Copyright terms: Public domain | W3C validator |