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Mirrors > Home > ILE Home > Th. List > mpteq2ia | GIF version |
Description: An equality inference for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpteq2ia.1 | ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
mpteq2ia | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | ax-gen 1425 | . 2 ⊢ ∀𝑥 𝐴 = 𝐴 |
3 | mpteq2ia.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | rgen 2485 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶 |
5 | mpteq12f 4008 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
6 | 2, 4, 5 | mp2an 422 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 = 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: mpteq2i 4015 feqresmpt 5475 elfvmptrab 5516 fmptap 5610 offres 6033 cnrecnv 10689 ege2le3 11384 eirraplem 11490 cnmpt1st 12467 cnmpt2nd 12468 expcncf 12771 dvexp 12854 dveflem 12865 dvef 12866 |
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