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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfateq12d | Structured version Visualization version GIF version |
Description: Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
dfateq12d.1 | ⊢ (𝜑 → 𝐹 = 𝐺) |
dfateq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
dfateq12d | ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfateq12d.2 | . . . 4 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | dfateq12d.1 | . . . . 5 ⊢ (𝜑 → 𝐹 = 𝐺) | |
3 | 2 | dmeqd 5767 | . . . 4 ⊢ (𝜑 → dom 𝐹 = dom 𝐺) |
4 | 1, 3 | eleq12d 2904 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐵 ∈ dom 𝐺)) |
5 | 1 | sneqd 4569 | . . . . 5 ⊢ (𝜑 → {𝐴} = {𝐵}) |
6 | 2, 5 | reseq12d 5847 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵})) |
7 | 6 | funeqd 6370 | . . 3 ⊢ (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵}))) |
8 | 4, 7 | anbi12d 630 | . 2 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))) |
9 | df-dfat 43195 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 43195 | . 2 ⊢ (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))) | |
11 | 8, 9, 10 | 3bitr4g 315 | 1 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {csn 4557 dom cdm 5548 ↾ cres 5550 Fun wfun 6342 defAt wdfat 43192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-res 5560 df-fun 6350 df-dfat 43195 |
This theorem is referenced by: afveq12d 43209 afv2eq12d 43291 |
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