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| Mirrors > Home > MPE Home > Th. List > mpoex | 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 Mario Carneiro, 20-Dec-2013.) |
| Ref | Expression |
|---|---|
| mpoex.1 | ⊢ 𝐴 ∈ V |
| mpoex.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| mpoex | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpoex.1 | . 2 ⊢ 𝐴 ∈ V | |
| 2 | mpoex.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 3 | 2 | rgenw 3065 | . 2 ⊢ ∀𝑥 ∈ 𝐴 𝐵 ∈ V |
| 4 | eqid 2737 | . . 3 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
| 5 | 4 | mpoexxg 8100 | . 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ V) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V) |
| 6 | 1, 3, 5 | mp2an 692 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ∈ V |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 ∀wral 3061 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: mptmpoopabbrdOLD 8106 qexALT 13006 ruclem13 16278 vdwapfval 17009 prdsco 17513 imasvsca 17565 homffval 17733 comfffval 17741 comffval 17742 comfffn 17747 comfeq 17749 oppccofval 17759 monfval 17776 sectffval 17794 invffval 17802 cofu1st 17928 cofu2nd 17930 cofucl 17933 natfval 17994 fuccofval 18007 fucco 18010 coafval 18109 setcco 18128 catchomfval 18147 catccofval 18149 catcco 18150 estrcco 18174 xpcval 18222 xpchomfval 18224 xpccofval 18227 xpcco 18228 1stf1 18237 1stf2 18238 2ndf1 18240 2ndf2 18241 1stfcl 18242 2ndfcl 18243 prf1 18245 prf2fval 18246 prfcl 18248 prf1st 18249 prf2nd 18250 evlf2 18263 evlf1 18265 evlfcl 18267 curf1fval 18269 curf11 18271 curf12 18272 curf1cl 18273 curf2 18274 curfcl 18277 hof1fval 18298 hof2fval 18300 hofcl 18304 yonedalem3 18325 efmndplusg 18893 mgmnsgrpex 18944 sgrpnmndex 18945 grpsubfvalALT 19002 mulgfvalALT 19088 symgvalstruct 19414 symgvalstructOLD 19415 lsmfval 19656 pj1fval 19712 dvrfval 20402 psrmulr 21962 psrvscafval 21968 evlslem2 22103 mamufval 22396 mvmulfval 22548 isphtpy 25013 pcofval 25043 q1pval 26194 r1pval 26197 mulsproplem9 28150 motplusg 28550 midf 28784 ismidb 28786 ttgval 28883 ttgvalOLD 28884 ebtwntg 28997 ecgrtg 28998 elntg 28999 wwlksnon 29871 wspthsnon 29872 clwwlknonmpo 30108 vsfval 30652 dipfval 30721 idlsrgmulr 33535 smatfval 33794 lmatval 33812 qqhval 33973 dya2iocuni 34285 sxbrsigalem5 34290 sitmval 34351 signswplusg 34570 reprval 34625 mclsrcl 35566 mclsval 35568 ldualfvs 39137 paddfval 39799 tgrpopr 40749 erngfplus 40804 erngfmul 40807 erngfplus-rN 40812 erngfmul-rN 40815 dvafvadd 41016 dvafvsca 41018 dvaabl 41026 dvhfvadd 41093 dvhfvsca 41102 djafvalN 41136 djhfval 41399 hlhilip 41954 mendplusgfval 43193 mendmulrfval 43195 mendvscafval 43198 mnringmulrd 44240 mnringmulrcld 44247 hoidmvval 46592 cznrng 48177 cznnring 48178 rngchomfvalALTV 48183 rngccofvalALTV 48186 rngccoALTV 48187 ringchomfvalALTV 48217 ringccofvalALTV 48220 ringccoALTV 48221 rrx2xpreen 48640 lines 48652 spheres 48667 funcf2lem2 48915 upfval 48933 swapfelvv 48969 swapf2fvala 48970 swapf1vala 48972 tposcurf1 48999 fucoelvv 49015 fucofn2 49019 fucofvalne 49020 fuco112 49024 fuco111 49025 fuco21 49031 functhinclem1 49093 thincciso 49102 functermc2 49141 |
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