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Mirrors > Home > MPE Home > Th. List > dmmpti | Structured version Visualization version GIF version |
Description: Domain of the mapping operation. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fnmpti.1 | ⊢ 𝐵 ∈ V |
fnmpti.2 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
dmmpti | ⊢ dom 𝐹 = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmpti.1 | . . 3 ⊢ 𝐵 ∈ V | |
2 | fnmpti.2 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | fnmpti 5935 | . 2 ⊢ 𝐹 Fn 𝐴 |
4 | fndm 5904 | . 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
5 | 3, 4 | ax-mp 5 | 1 ⊢ dom 𝐹 = 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 dom cdm 5038 Fn wfn 5799 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
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-ral 2901 df-rab 2905 df-v 3175 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-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-fun 5806 df-fn 5807 |
This theorem is referenced by: fvmptex 6203 resfunexg 6384 brtpos2 7245 vdwlem8 15530 lubdm 16802 glbdm 16815 dprd2dlem2 18262 dprd2dlem1 18263 dprd2da 18264 ablfac1c 18293 ablfac1eu 18295 ablfaclem2 18308 ablfaclem3 18309 elocv 19831 dfac14 21231 kqtop 21358 symgtgp 21715 eltsms 21746 ressprdsds 21986 minveclem1 23003 isi1f 23247 itg1val 23256 cmvth 23558 mvth 23559 lhop2 23582 dvfsumabs 23590 dvfsumrlim2 23599 taylthlem1 23931 taylthlem2 23932 ulmdvlem1 23958 pige3 24073 relogcn 24184 atandm 24403 atanf 24407 atancn 24463 dmarea 24484 dfarea 24487 efrlim 24496 lgamgulmlem2 24556 dchrptlem2 24790 dchrptlem3 24791 dchrisum0 25009 eleenn 25576 incistruhgr 25746 vsfval 26872 ipasslem8 27076 minvecolem1 27114 xppreima2 28830 ofpreima 28848 dmsigagen 29534 measbase 29587 sseqf 29781 ballotlem7 29924 bj-dmtopon 32242 bj-inftyexpidisj 32274 bj-elccinfty 32278 bj-minftyccb 32289 fin2so 32566 poimirlem30 32609 poimir 32612 dvtan 32630 itg2addnclem2 32632 ftc1anclem6 32660 totbndbnd 32758 comptiunov2i 37017 lhe4.4ex1a 37550 dvsinax 38801 fourierdlem62 39061 fourierdlem70 39069 fourierdlem71 39070 fourierdlem80 39079 fouriersw 39124 mndpsuppss 41946 scmsuppss 41947 lincext2 42038 aacllem 42356 |
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