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Mirrors > Home > ILE Home > Th. List > mpteq1 | GIF version |
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq1 | ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2140 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐶 = 𝐶) | |
2 | 1 | rgen 2485 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶 |
3 | mpteq12 4011 | . 2 ⊢ ((𝐴 = 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝐶 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) | |
4 | 2, 3 | mpan2 421 | 1 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑥 ∈ 𝐵 ↦ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ↦ 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: mpteq1d 4013 fmptap 5610 mpompt 5863 mpomptsx 6095 mpompts 6096 tposf12 6166 restco 12343 cnmpt1t 12454 cnmpt2t 12462 |
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