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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 5248 | . . . 4 ⊢ (𝜑 → dom 𝐹 = dom 𝐺) |
4 | 1, 3 | eleq12d 2682 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐵 ∈ dom 𝐺)) |
5 | 1 | sneqd 4137 | . . . . 5 ⊢ (𝜑 → {𝐴} = {𝐵}) |
6 | 2, 5 | reseq12d 5318 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵})) |
7 | 6 | funeqd 5825 | . . 3 ⊢ (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵}))) |
8 | 4, 7 | anbi12d 743 | . 2 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))) |
9 | df-dfat 39845 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 39845 | . 2 ⊢ (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))) | |
11 | 8, 9, 10 | 3bitr4g 302 | 1 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {csn 4125 dom cdm 5038 ↾ cres 5040 Fun wfun 5798 defAt wdfat 39842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-res 5050 df-fun 5806 df-dfat 39845 |
This theorem is referenced by: afveq12d 39862 |
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