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Mirrors > Home > MPE Home > Th. List > fnmpoi | Structured version Visualization version GIF version |
Description: Functionality and domain of a class given by the maps-to notation. (Contributed by FL, 17-May-2010.) |
Ref | Expression |
---|---|
fmpo.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
fnmpoi.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
fnmpoi | ⊢ 𝐹 Fn (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmpoi.2 | . . 3 ⊢ 𝐶 ∈ V | |
2 | 1 | rgen2w 3151 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ V |
3 | fmpo.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
4 | 3 | fnmpo 7767 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ V → 𝐹 Fn (𝐴 × 𝐵)) |
5 | 2, 4 | ax-mp 5 | 1 ⊢ 𝐹 Fn (𝐴 × 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 × cxp 5553 Fn wfn 6350 ∈ 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-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-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-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-fv 6363 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: dmmpo 7769 fnoa 8133 fnom 8134 fnoe 8135 fnmap 8413 fnpm 8414 addpqnq 10360 mulpqnq 10363 elq 12351 cnref1o 12385 ccatfn 13924 qnnen 15566 restfn 16698 prdsdsfn 16738 imasdsfn 16787 imasvscafn 16810 homffn 16963 comfffn 16974 comffn 16975 isoval 17035 cofucl 17158 fnfuc 17215 natffn 17219 catcisolem 17366 estrchomfn 17385 funcestrcsetclem4 17393 funcsetcestrclem4 17408 fnxpc 17426 1stfcl 17447 2ndfcl 17448 prfcl 17453 evlfcl 17472 curf1cl 17478 curfcl 17482 hofcl 17509 yonedalem3 17530 yonedainv 17531 plusffn 17861 mulgfval 18226 mulgfvalALT 18227 mulgfn 18229 gimfn 18401 sylow2blem2 18746 scaffn 19655 lmimfn 19798 mplsubrglem 20219 ipffn 20795 tx1stc 22258 tx2ndc 22259 hmeofn 22365 efmndtmd 22709 qustgplem 22729 nmoffn 23320 rrxmfval 24009 mbfimaopnlem 24256 i1fadd 24296 i1fmul 24297 ex-fpar 28241 smatrcl 31061 txomap 31098 qtophaus 31100 pstmxmet 31137 dya2icoseg 31535 dya2iocrfn 31537 fncvm 32504 cntotbnd 35089 rnghmfn 44181 rhmfn 44209 rnghmsscmap2 44264 rnghmsscmap 44265 rngchomffvalALTV 44286 rngchomrnghmresALTV 44287 rhmsscmap2 44310 rhmsscmap 44311 funcringcsetcALTV2lem4 44330 funcringcsetclem4ALTV 44353 srhmsubc 44367 fldc 44374 fldhmsubc 44375 rhmsubclem1 44377 srhmsubcALTV 44385 fldcALTV 44392 fldhmsubcALTV 44393 rhmsubcALTVlem1 44395 rrx2xpref1o 44725 |
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