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| Mirrors > Home > MPE Home > Th. List > mpoexga | Structured version Visualization version GIF version | ||
| Description: If the domain of an operation given by maps-to notation is a set, the operation is a set. (Contributed by NM, 12-Sep-2011.) |
| Ref | Expression |
|---|---|
| mpoexga | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
| 2 | 1 | mpoexg 8101 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 ∈ cmpo 7433 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: mptmpoopabbrdOLDOLD 8108 el2mpocsbcl 8110 bropopvvv 8115 bropfvvvv 8117 prdsip 17506 imasds 17558 isofn 17819 setchomfval 18124 setccofval 18127 estrchomfval 18170 estrccofval 18173 lsmvalx 19657 dfrngc2 20628 funcrngcsetc 20640 dfringc2 20657 funcringcsetc 20674 mamuval 22397 mamudm 22399 marrepfval 22566 marrepval0 22567 marrepval 22568 marepvfval 22571 marepvval 22573 submaval0 22586 submaval 22587 maduval 22644 minmar1val0 22653 minmar1val 22654 mat2pmatval 22730 mat2pmatf 22734 m2cpmf 22748 cpm2mval 22756 decpmatval0 22770 decpmatmul 22778 pmatcollpw2lem 22783 pmatcollpw3lem 22789 mply1topmatval 22810 mp2pm2mplem1 22812 xkoptsub 23662 precsexlem11 28241 grpodivfval 30553 pstmval 33894 sxsigon 34193 cndprobval 34435 lmod1lem1 48404 lmod1lem2 48405 lmod1lem3 48406 lmod1lem4 48407 lmod1lem5 48408 2arymaptfv 48572 2arymaptfo 48575 |
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