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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 2189 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | ax-gen 1460 | . 2 ⊢ ∀𝑥 𝐴 = 𝐴 |
3 | mpteq2ia.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐶) | |
4 | 3 | rgen 2543 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶 |
5 | mpteq12f 4098 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐶) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶)) | |
6 | 2, 4, 5 | mp2an 426 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1362 = wceq 1364 ∈ wcel 2160 ∀wral 2468 ↦ cmpt 4079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-ral 2473 df-opab 4080 df-mpt 4081 |
This theorem is referenced by: mpteq2i 4105 feqresmpt 5591 elfvmptrab 5632 fmptap 5727 offres 6160 cnrecnv 10951 ege2le3 11711 eirraplem 11816 cnmpt1st 14245 cnmpt2nd 14246 expcncf 14549 dvexp 14632 dveflem 14644 dvef 14645 |
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