![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfmpt3 | Structured version Visualization version GIF version |
Description: Alternate definition for the maps-to notation df-mpt 5233. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
dfmpt3 | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 5233 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | velsn 4646 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
3 | 2 | anbi2i 621 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
4 | 3 | anbi2i 621 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) |
5 | 4 | 2exbii 1843 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) |
6 | eliunxp 5840 | . . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}))) | |
7 | elopab 5529 | . . . 4 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) | |
8 | 5, 6, 7 | 3bitr4i 302 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}) |
9 | 8 | eqriv 2722 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
10 | 1, 9 | eqtr4i 2756 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {csn 4630 〈cop 4636 ∪ ciun 4997 {copab 5211 ↦ cmpt 5232 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-iun 4999 df-opab 5212 df-mpt 5233 df-xp 5684 df-rel 5685 |
This theorem is referenced by: dfmpt 7153 taylpfval 26344 indval2 33764 |
Copyright terms: Public domain | W3C validator |