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Mirrors > Home > MPE Home > Th. List > dfmpt3 | Structured version Visualization version GIF version |
Description: Alternate definition for the maps-to notation df-mpt 5042. (Contributed by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
dfmpt3 | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-mpt 5042 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | velsn 4488 | . . . . . . 7 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵) | |
3 | 2 | anbi2i 622 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) |
4 | 3 | anbi2i 622 | . . . . 5 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) |
5 | 4 | 2exbii 1830 | . . . 4 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵})) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) |
6 | eliunxp 5594 | . . . 4 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ {𝐵}))) | |
7 | elopab 5304 | . . . 4 ⊢ (𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))) | |
8 | 5, 6, 7 | 3bitr4i 304 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) ↔ 𝑧 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}) |
9 | 8 | eqriv 2792 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
10 | 1, 9 | eqtr4i 2822 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 {csn 4472 〈cop 4478 ∪ ciun 4825 {copab 5024 ↦ cmpt 5041 × cxp 5441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-iun 4827 df-opab 5025 df-mpt 5042 df-xp 5449 df-rel 5450 |
This theorem is referenced by: dfmpt 6769 taylpfval 24636 indval2 30890 |
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