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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 2081 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | ax-gen 1378 | . 2 ⊢ ∀𝑥 𝐴 = 𝐴 |
3 | mpteq2ia.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | rgen 2416 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶 |
5 | mpteq12f 3858 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
6 | 2, 4, 5 | mp2an 416 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1282 = wceq 1284 ∈ wcel 1433 ∀wral 2348 ↦ cmpt 3839 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-11 1437 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-ral 2353 df-opab 3840 df-mpt 3841 |
This theorem is referenced by: mpteq2i 3865 feqresmpt 5248 fmptap 5374 offres 5782 cnrecnv 9797 |
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