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Mirrors > Home > ILE Home > Th. List > dfmpt3 | GIF version |
Description: Alternate definition for the maps-to notation df-mpt 4063. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
dfmpt3 | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 4063 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | velsn 3608 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
3 | 2 | anbi2i 457 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
4 | 3 | anbi2i 457 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) |
5 | 4 | 2exbii 1606 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) |
6 | eliunxp 4762 | . . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}))) | |
7 | elopab 4255 | . . . 4 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) | |
8 | 5, 6, 7 | 3bitr4i 212 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}) |
9 | 8 | eqriv 2174 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
10 | 1, 9 | eqtr4i 2201 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∃wex 1492 ∈ wcel 2148 {csn 3591 〈cop 3594 ∪ ciun 3884 {copab 4060 ↦ cmpt 4061 × cxp 4621 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-iun 3886 df-opab 4062 df-mpt 4063 df-xp 4629 df-rel 4630 |
This theorem is referenced by: dfmpt 5689 dfmptg 5691 |
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