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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 4730 (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 6014 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) | |
2 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | dfmpt.1 | . . . . 5 ⊢ 𝐵 ∈ V | |
4 | 2, 3 | xpsn 6407 | . . . 4 ⊢ ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉} |
5 | 4 | a1i 11 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
6 | 5 | iuneq2i 4539 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
7 | 1, 6 | eqtri 2644 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 〈cop 4183 ∪ ciun 4520 ↦ cmpt 4729 × cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 |
This theorem is referenced by: fnasrn 6411 funiun 6412 dfmpt2 7267 |
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