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Mirrors > Home > ILE Home > Th. List > mpteq12dva | GIF version |
Description: An equality inference for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
Ref | Expression |
---|---|
mpteq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12dva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12dva | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12dv.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | 1 | alrimiv 1754 | . 2 ⊢ (𝜑 → ∀𝑥 𝐴 = 𝐶) |
3 | mpteq12dva.2 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
4 | 3 | ralrimiva 2392 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) |
5 | mpteq12f 3837 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | |
6 | 2, 4, 5 | syl2anc 391 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1241 = wceq 1243 ∈ wcel 1393 ∀wral 2306 ↦ cmpt 3818 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-ral 2311 df-opab 3819 df-mpt 3820 |
This theorem is referenced by: mpteq12dv 3839 |
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