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Mirrors > Home > ILE Home > Th. List > mpteq2da | GIF version |
Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013.) (Revised by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq2da.1 | ⊢ Ⅎ𝑥𝜑 |
mpteq2da.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
mpteq2da | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2137 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | ax-gen 1425 | . 2 ⊢ ∀𝑥 𝐴 = 𝐴 |
3 | mpteq2da.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
4 | mpteq2da.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐶) | |
5 | 4 | ex 114 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 = 𝐶)) |
6 | 3, 5 | ralrimi 2501 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
7 | mpteq12f 4003 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
8 | 2, 6, 7 | sylancr 410 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1329 = wceq 1331 Ⅎwnf 1436 ∈ wcel 1480 ∀wral 2414 ↦ cmpt 3984 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-ral 2419 df-opab 3985 df-mpt 3986 |
This theorem is referenced by: mpteq2dva 4013 prodeq1f 11314 prodeq2 11319 |
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