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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 5916 | . . . 4 ⊢ (𝜑 → dom 𝐹 = dom 𝐺) |
| 4 | 1, 3 | eleq12d 2835 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐵 ∈ dom 𝐺)) |
| 5 | 1 | sneqd 4638 | . . . . 5 ⊢ (𝜑 → {𝐴} = {𝐵}) |
| 6 | 2, 5 | reseq12d 5998 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵})) |
| 7 | 6 | funeqd 6588 | . . 3 ⊢ (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵}))) |
| 8 | 4, 7 | anbi12d 632 | . 2 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))) |
| 9 | df-dfat 47131 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
| 10 | df-dfat 47131 | . 2 ⊢ (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))) | |
| 11 | 8, 9, 10 | 3bitr4g 314 | 1 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 defAt wdfat 47128 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-fun 6563 df-dfat 47131 |
| This theorem is referenced by: afveq12d 47145 afv2eq12d 47227 |
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