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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 276 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) |
4 | 1, 3 | mpteq12dva 4081 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 ↦ cmpt 4061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-ral 2460 df-opab 4062 df-mpt 4063 |
This theorem is referenced by: mpteq12i 4088 offval 6083 offval3 6128 odzval 12211 restval 12629 grpinvfvalg 12792 grpinvpropdg 12821 opprnegg 13065 ntrfval 13233 clsfval 13234 neifval 13273 cnpfval 13328 cnprcl2k 13339 reldvg 13781 dvfvalap 13783 eldvap 13784 |
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