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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 2821 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
2 | 1 | mpoexg 7774 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 Vcvv 3494 ∈ cmpo 7158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: mptmpoopabbrd 7778 el2mpocsbcl 7780 bropopvvv 7785 bropfvvvv 7787 prdsip 16734 imasds 16786 isofn 17045 setchomfval 17339 setccofval 17342 estrchomfval 17376 estrccofval 17379 lsmvalx 18764 mamuval 20997 mamudm 20999 marrepfval 21169 marrepval0 21170 marrepval 21171 marepvfval 21174 marepvval 21176 submaval0 21189 submaval 21190 maduval 21247 minmar1val0 21256 minmar1val 21257 mat2pmatval 21332 mat2pmatf 21336 m2cpmf 21350 cpm2mval 21358 decpmatval0 21372 decpmatmul 21380 pmatcollpw2lem 21385 pmatcollpw3lem 21391 mply1topmatval 21412 mp2pm2mplem1 21414 xkoptsub 22262 grpodivfval 28311 pstmval 31135 sxsigon 31451 cndprobval 31691 dfrngc2 44292 funcrngcsetc 44318 dfringc2 44338 funcringcsetc 44355 lmod1lem1 44591 lmod1lem2 44592 lmod1lem3 44593 lmod1lem4 44594 lmod1lem5 44595 |
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