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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 5232 (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 6689 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) | |
2 | vex 3475 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | dfmpt.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | xpsn 7150 | . . . 4 ⊢ ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩} |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝐵}) = {⟨𝑥, 𝐵⟩}) |
6 | 5 | iuneq2i 5017 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} |
7 | 1, 6 | eqtri 2756 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {⟨𝑥, 𝐵⟩} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 {csn 4629 ⟨cop 4635 ∪ ciun 4996 ↦ cmpt 5231 × cxp 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 |
This theorem is referenced by: fnasrn 7154 funiun 7156 dfmpo 8107 |
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