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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 5738 | . . . 4 ⊢ (𝜑 → dom 𝐹 = dom 𝐺) |
4 | 1, 3 | eleq12d 2884 | . . 3 ⊢ (𝜑 → (𝐴 ∈ dom 𝐹 ↔ 𝐵 ∈ dom 𝐺)) |
5 | 1 | sneqd 4537 | . . . . 5 ⊢ (𝜑 → {𝐴} = {𝐵}) |
6 | 2, 5 | reseq12d 5819 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ {𝐴}) = (𝐺 ↾ {𝐵})) |
7 | 6 | funeqd 6346 | . . 3 ⊢ (𝜑 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun (𝐺 ↾ {𝐵}))) |
8 | 4, 7 | anbi12d 633 | . 2 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵})))) |
9 | df-dfat 43675 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
10 | df-dfat 43675 | . 2 ⊢ (𝐺 defAt 𝐵 ↔ (𝐵 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐵}))) | |
11 | 8, 9, 10 | 3bitr4g 317 | 1 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {csn 4525 dom cdm 5519 ↾ cres 5521 Fun wfun 6318 defAt wdfat 43672 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-fun 6326 df-dfat 43675 |
This theorem is referenced by: afveq12d 43689 afv2eq12d 43771 |
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