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Mirrors > Home > ILE Home > Th. List > mpteq12 | GIF version |
Description: An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq12 | ⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1491 | . 2 ⊢ (𝐴 = 𝐶 → ∀𝑥 𝐴 = 𝐶) | |
2 | mpteq12f 3978 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) | |
3 | 1, 2 | sylan 281 | 1 ⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1314 = wceq 1316 ∀wral 2393 ↦ cmpt 3959 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-opab 3960 df-mpt 3961 |
This theorem is referenced by: mpteq1 3982 mpteqb 5479 fmptcof 5555 mapxpen 6710 |
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