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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 5881 | . . . 4 ⊢ (𝜑 → dom 𝐹 = dom 𝐺) |
4 | 1, 3 | eleq12d 2826 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐵 ∈ dom 𝐺)) |
5 | 1 | sneqd 4618 | . . . . 5 ⊢ (𝜑 → {𝐴} = {𝐵}) |
6 | 2, 5 | reseq12d 5958 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵})) |
7 | 6 | funeqd 6543 | . . 3 ⊢ (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵}))) |
8 | 4, 7 | anbi12d 631 | . 2 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))) |
9 | df-dfat 45504 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 45504 | . 2 ⊢ (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))) | |
11 | 8, 9, 10 | 3bitr4g 313 | 1 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 {csn 4606 dom cdm 5653 ↾ cres 5655 Fun wfun 6510 defAt wdfat 45501 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-br 5126 df-opab 5188 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-res 5665 df-fun 6518 df-dfat 45504 |
This theorem is referenced by: afveq12d 45518 afv2eq12d 45600 |
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