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Mirrors > Home > MPE Home > Th. List > dfmpt | Structured version Visualization version GIF version |
Description: Alternate definition for the maps-to notation df-mpt 5223 (although it requires that 𝐵 be a set). (Contributed by NM, 24-Aug-2010.) (Revised by Mario Carneiro, 30-Dec-2016.) |
Ref | Expression |
---|---|
dfmpt.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
dfmpt | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfmpt3 6675 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) | |
2 | vex 3470 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | dfmpt.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | xpsn 7132 | . . . 4 ⊢ ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩} |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}) |
6 | 5 | iuneq2i 5009 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} |
7 | 1, 6 | eqtri 2752 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3466 {csn 4621 ⟨cop 4627 ∪ ciun 4988 ↦ cmpt 5222 × cxp 5665 |
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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 |
This theorem is referenced by: fnasrn 7136 funiun 7138 dfmpo 8083 |
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