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| Mirrors > Home > MPE Home > Th. List > fnmpt | Structured version Visualization version GIF version | ||
| Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) |
| Ref | Expression |
|---|---|
| mptfng.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| Ref | Expression |
|---|---|
| fnmpt | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
| 2 | 1 | ralimi 3083 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
| 3 | mptfng.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 4 | 3 | mptfng 6707 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴) |
| 5 | 2, 4 | sylib 218 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ↦ cmpt 5225 Fn wfn 6556 |
| 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 |
| 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-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 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-fun 6563 df-fn 6564 |
| This theorem is referenced by: fnmptd 6709 mpt0 6710 fnmptfvd 7061 ralrnmptw 7114 ralrnmpt 7116 fmpt 7130 fmpt2d 7144 f1ocnvd 7684 offval2 7717 ofrfval2 7718 mptcnfimad 8011 fsplitfpar 8143 mptelixpg 8975 fifo 9472 cantnflem1 9729 infmap2 10257 compssiso 10414 gruiun 10839 mptnn0fsupp 14038 mptnn0fsuppr 14040 seqof 14100 rlimi2 15550 prdsbas3 17526 prdsbascl 17528 prdsdsval2 17529 quslem 17588 fnmrc 17650 isofn 17819 ghmquskerco 19302 pmtrrn 19475 pmtrfrn 19476 pmtrdifwrdellem2 19500 gsummptcl 19985 mptscmfsupp0 20925 ofco2 22457 pmatcollpw2lem 22783 neif 23108 tgrest 23167 cmpfi 23416 elptr2 23582 xkoptsub 23662 ptcmplem2 24061 ptcmplem3 24062 prdsxmetlem 24378 prdsxmslem2 24542 bcth3 25365 uniioombllem6 25623 itg2const 25775 ellimc2 25912 dvrec 25993 dvmptres3 25994 ulmss 26440 ulmdvlem1 26443 ulmdvlem2 26444 ulmdvlem3 26445 itgulm2 26452 psercn 26470 tgjustr 28482 f1o3d 32637 f1od2 32732 psgnfzto1stlem 33120 frlmdim 33662 rmulccn 33927 esumnul 34049 esum0 34050 gsumesum 34060 ofcfval2 34105 signsplypnf 34565 signsply0 34566 hgt750lemb 34671 wevgblacfn 35114 matunitlindflem1 37623 matunitlindflem2 37624 cdlemk56 40973 dicfnN 41185 hbtlem7 43137 refsumcn 45035 wessf1ornlem 45190 choicefi 45205 axccdom 45227 fsumsermpt 45594 liminfval2 45783 stoweidlem31 46046 stoweidlem59 46074 stirlinglem13 46101 dirkercncflem2 46119 fourierdlem62 46183 subsaliuncllem 46372 subsaliuncl 46373 hoidmvlelem3 46612 dfafn5b 47173 fundcmpsurinjlem2 47386 lincresunit2 48395 |
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