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Mirrors > Home > ILE Home > Th. List > mpteq12i | GIF version |
Description: An equality inference for the maps-to notation. (Contributed by Scott Fenton, 27-Oct-2010.) (Revised by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq12i.1 | ⊢ 𝐴 = 𝐶 |
mpteq12i.2 | ⊢ 𝐵 = 𝐷 |
Ref | Expression |
---|---|
mpteq12i | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12i.1 | . . . 4 ⊢ 𝐴 = 𝐶 | |
2 | 1 | a1i 9 | . . 3 ⊢ (⊤ → 𝐴 = 𝐶) |
3 | mpteq12i.2 | . . . 4 ⊢ 𝐵 = 𝐷 | |
4 | 3 | a1i 9 | . . 3 ⊢ (⊤ → 𝐵 = 𝐷) |
5 | 2, 4 | mpteq12dv 4058 | . 2 ⊢ (⊤ → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
6 | 5 | mptru 1351 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ⊤wtru 1343 ↦ 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: offres 6095 |
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