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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 3066 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ V |
| 3 | fmpo.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
| 4 | 3 | fnmpo 8094 | . 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 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 × cxp 5683 Fn wfn 6556 ∈ 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-sep 5296 ax-nul 5306 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-ral 3062 df-rex 3071 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-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-fv 6569 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 |
| This theorem is referenced by: dmmpo 8096 fnoa 8546 fnom 8547 fnoe 8548 fnmap 8873 fnpm 8874 addpqnq 10978 mulpqnq 10981 mpoaddf 11249 mpomulf 11250 elq 12992 cnref1o 13027 ccatfn 14610 qnnen 16249 restfn 17469 prdsdsfn 17510 imasdsfn 17559 imasvscafn 17582 homffn 17736 comfffn 17747 comffn 17748 isoval 17809 cofucl 17933 fnfuc 17993 natffn 17997 catcisolem 18155 estrchomfn 18179 funcestrcsetclem4 18188 funcsetcestrclem4 18203 fnxpc 18221 1stfcl 18242 2ndfcl 18243 prfcl 18248 evlfcl 18267 curf1cl 18273 curfcl 18277 hofcl 18304 yonedalem3 18325 yonedainv 18326 plusffn 18662 mulgfval 19087 mulgfvalALT 19088 mulgfn 19090 gimfn 19279 sylow2blem2 19639 rnghmfn 20439 rhmfn 20499 rnghmsscmap2 20629 rnghmsscmap 20630 rhmsscmap2 20658 rhmsscmap 20659 srhmsubc 20680 rhmsubclem1 20685 fldc 20785 fldhmsubc 20786 scaffn 20881 lmimfn 21025 ipffn 21669 mplsubrglem 22024 tx1stc 23658 tx2ndc 23659 hmeofn 23765 efmndtmd 24109 qustgplem 24129 nmoffn 24732 rrxmfval 25440 mbfimaopnlem 25690 i1fadd 25730 i1fmul 25731 subsfn 28056 ex-fpar 30481 smatrcl 33795 txomap 33833 qtophaus 33835 pstmxmet 33896 dya2icoseg 34279 dya2iocrfn 34281 fncvm 35262 mpomulnzcnf 36300 cntotbnd 37803 grimfn 47865 grlimfn 47946 rngchomffvalALTV 48194 rngchomrnghmresALTV 48195 rhmsubcALTVlem1 48197 funcringcsetcALTV2lem4 48209 funcringcsetclem4ALTV 48232 srhmsubcALTV 48241 fldcALTV 48248 fldhmsubcALTV 48249 rrx2xpref1o 48639 swapf2fn 48974 fucofn2 49019 functhinclem1 49093 |
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