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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 4057 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 ↦ 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: mpteq12i 4064 offval 6051 offval3 6094 odzval 12152 restval 12504 ntrfval 12647 clsfval 12648 neifval 12687 cnpfval 12742 cnprcl2k 12753 reldvg 13195 dvfvalap 13197 eldvap 13198 |
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