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Mirrors > Home > ILE Home > Th. List > mpoeq12 | GIF version |
Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
Ref | Expression |
---|---|
mpoeq12 | ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2140 | . . . . 5 ⊢ 𝐸 = 𝐸 | |
2 | 1 | rgenw 2490 | . . . 4 ⊢ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸 |
3 | 2 | jctr 313 | . . 3 ⊢ (𝐵 = 𝐷 → (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸)) |
4 | 3 | ralrimivw 2509 | . 2 ⊢ (𝐵 = 𝐷 → ∀𝑥 ∈ 𝐴 (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸)) |
5 | mpoeq123 5838 | . 2 ⊢ ((𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 (𝐵 = 𝐷 ∧ ∀𝑦 ∈ 𝐵 𝐸 = 𝐸)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) | |
6 | 4, 5 | sylan2 284 | 1 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐸) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝐸)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∀wral 2417 ∈ cmpo 5784 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-oprab 5786 df-mpo 5787 |
This theorem is referenced by: seqeq1 10252 txvalex 12462 txval 12463 |
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