Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6493 | . 2 ⊢ 𝐹 Fn 𝐴 |
4 | fndm 6457 | . 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 1537 ∈ wcel 2114 Vcvv 3496 ↦ cmpt 5148 dom cdm 5557 Fn wfn 6352 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-fun 6359 df-fn 6360 |
This theorem is referenced by: fvmptex 6784 resfunexg 6980 brtpos2 7900 inlresf 9345 inrresf 9347 vdwlem8 16326 lubdm 17591 glbdm 17604 dprd2dlem2 19164 dprd2dlem1 19165 dprd2da 19166 ablfac1c 19195 ablfac1eu 19197 ablfaclem2 19210 ablfaclem3 19211 elocv 20814 dmtopon 21533 dfac14 22228 kqtop 22355 symgtgp 22716 eltsms 22743 ressprdsds 22983 minveclem1 24029 isi1f 24277 itg1val 24286 cmvth 24590 mvth 24591 lhop2 24614 dvfsumabs 24622 dvfsumrlim2 24631 taylthlem1 24963 taylthlem2 24964 ulmdvlem1 24990 pige3ALT 25107 relogcn 25223 atandm 25456 atanf 25460 atancn 25516 dmarea 25537 dfarea 25540 efrlim 25549 lgamgulmlem2 25609 dchrptlem2 25843 dchrptlem3 25844 dchrisum0 26098 incistruhgr 26866 vsfval 28412 ipasslem8 28616 minvecolem1 28653 xppreima2 30397 ofpreima 30412 rmfsupp2 30868 dmsigagen 31405 measbase 31458 sseqf 31652 ballotlem7 31795 nosupno 33205 nosupdm 33206 nosupbday 33207 nosupres 33209 nosupbnd1lem1 33210 bj-inftyexpitaudisj 34489 bj-inftyexpidisj 34494 bj-elccinfty 34498 bj-minftyccb 34509 fin2so 34881 poimirlem30 34924 poimir 34927 dvtan 34944 itg2addnclem2 34946 ftc1anclem6 34974 totbndbnd 35069 comptiunov2i 40058 lhe4.4ex1a 40668 dvsinax 42204 fourierdlem62 42460 fourierdlem70 42468 fourierdlem71 42469 fourierdlem80 42478 fouriersw 42523 smflimsuplem1 43101 smflimsuplem4 43104 mndpsuppss 44426 scmsuppss 44427 lincext2 44517 aacllem 44909 |
Copyright terms: Public domain | W3C validator |