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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 4645 (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 5927 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) | |
2 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | dfmpt.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | xpsn 6313 | . . . 4 ⊢ ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉} |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
6 | 5 | iuneq2i 4475 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
7 | 1, 6 | eqtri 2632 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 ∪ ciun 4455 ↦ cmpt 4643 × cxp 5036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 |
This theorem is referenced by: fnasrn 6317 funiun 6318 dfmpt2 7154 |
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