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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 2164 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | ax-gen 1436 | . 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 2535 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) |
7 | mpteq12f 4056 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
8 | 2, 6, 7 | sylancr 411 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1340 = wceq 1342 Ⅎwnf 1447 ∈ wcel 2135 ∀wral 2442 ↦ cmpt 4037 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-11 1493 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-ral 2447 df-opab 4038 df-mpt 4039 |
This theorem is referenced by: mpteq2dva 4066 prodeq1f 11479 prodeq2 11484 |
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