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Mirrors > Home > MPE Home > Th. List > fnmpt2i | 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 |
---|---|
fmpt2.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) |
fnmpt2i.2 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
fnmpt2i | ⊢ 𝐹 Fn (𝐴 × 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmpt2i.2 | . . 3 ⊢ 𝐶 ∈ V | |
2 | 1 | rgen2w 2909 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝐶 ∈ V |
3 | fmpt2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) | |
4 | 3 | fnmpt2 7127 | . 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 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 × cxp 5036 Fn wfn 5799 ↦ cmpt2 6551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 |
This theorem is referenced by: dmmpt2 7129 fnoa 7475 fnom 7476 fnoe 7477 fnmap 7751 fnpm 7752 cdafn 8874 addpqnq 9639 mulpqnq 9642 elq 11666 cnref1o 11703 ccatfn 13210 qnnen 14781 restfn 15908 prdsdsfn 15948 imasdsfn 15997 imasvscafn 16020 homffn 16176 comfffn 16187 comffn 16188 isoval 16248 cofucl 16371 fnfuc 16428 natffn 16432 catcisolem 16579 estrchomfn 16598 funcestrcsetclem4 16606 funcsetcestrclem4 16621 fnxpc 16639 1stfcl 16660 2ndfcl 16661 prfcl 16666 evlfcl 16685 curf1cl 16691 curfcl 16695 hofcl 16722 yonedalem3 16743 yonedainv 16744 plusffn 17073 mulgfval 17365 mulgfn 17367 gimfn 17526 symgplusg 17632 sylow2blem2 17859 scaffn 18707 lmimfn 18847 mplsubrglem 19260 ipffn 19815 tx1stc 21263 tx2ndc 21264 hmeofn 21370 symgtgp 21715 qustgplem 21734 nmoffn 22325 rrxmfval 22997 mbfimaopnlem 23228 i1fadd 23268 i1fmul 23269 smatrcl 29190 txomap 29229 qtophaus 29231 pstmxmet 29268 dya2icoseg 29666 dya2iocrfn 29668 fncvm 30493 cntotbnd 32765 rnghmfn 41680 rhmfn 41708 rnghmsscmap2 41765 rnghmsscmap 41766 rngchomffvalALTV 41787 rngchomrnghmresALTV 41788 rhmsscmap2 41811 rhmsscmap 41812 funcringcsetcALTV2lem4 41831 funcringcsetclem4ALTV 41854 srhmsubc 41868 fldc 41875 fldhmsubc 41876 rhmsubclem1 41878 srhmsubcALTV 41887 fldcALTV 41894 fldhmsubcALTV 41895 rhmsubcALTVlem1 41897 |
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