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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 6444. (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 42022 | . . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
4 | eqidd 2826 | . . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥) | |
5 | 2, 1, 4 | breq123d 4889 | . . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥)) |
6 | 5 | iotabidv 6111 | . . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥)) |
7 | 1 | rneqd 5589 | . . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺) |
8 | 7 | unieqd 4670 | . . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺) |
9 | 8 | pweqd 4385 | . . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺) |
10 | 3, 6, 9 | ifbieq12d 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) | |
13 | 10, 11, 12 | 3eqtr4g 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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