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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 8010 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 Vcvv 3446 ∈ cmpo 7360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 |
This theorem is referenced by: mptmpoopabbrdOLD 8016 el2mpocsbcl 8018 bropopvvv 8023 bropfvvvv 8025 prdsip 17344 imasds 17396 isofn 17659 setchomfval 17966 setccofval 17969 estrchomfval 18014 estrccofval 18017 lsmvalx 19422 mamuval 21738 mamudm 21740 marrepfval 21912 marrepval0 21913 marrepval 21914 marepvfval 21917 marepvval 21919 submaval0 21932 submaval 21933 maduval 21990 minmar1val0 21999 minmar1val 22000 mat2pmatval 22076 mat2pmatf 22080 m2cpmf 22094 cpm2mval 22102 decpmatval0 22116 decpmatmul 22124 pmatcollpw2lem 22129 pmatcollpw3lem 22135 mply1topmatval 22156 mp2pm2mplem1 22158 xkoptsub 23008 grpodivfval 29479 pstmval 32479 sxsigon 32794 cndprobval 33036 dfrngc2 46277 funcrngcsetc 46303 dfringc2 46323 funcringcsetc 46340 lmod1lem1 46575 lmod1lem2 46576 lmod1lem3 46577 lmod1lem4 46578 lmod1lem5 46579 2arymaptfv 46744 2arymaptfo 46747 |
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