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Mirrors > Home > ILE Home > Th. List > mpteq12dv | GIF version |
Description: An equality inference for the maps-to notation. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq12dv.1 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12dv.2 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12dv | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12dv.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | mpteq12dv.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐷) | |
3 | 2 | adantr 274 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
4 | 1, 3 | mpteq12dva 4009 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ↦ cmpt 3989 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-ral 2421 df-opab 3990 df-mpt 3991 |
This theorem is referenced by: mpteq12i 4016 offval 5989 offval3 6032 restval 12126 ntrfval 12269 clsfval 12270 neifval 12309 cnpfval 12364 cnprcl2k 12375 reldvg 12817 dvfvalap 12819 eldvap 12820 |
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