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Mirrors > Home > ILE Home > Th. List > fvmptg | Unicode version |
Description: Value of a function given in maps-to notation. (Contributed by NM, 2-Oct-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
fvmptg.1 | |
fvmptg.2 |
Ref | Expression |
---|---|
fvmptg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . 2 | |
2 | fvmptg.1 | . . . 4 | |
3 | 2 | eqeq2d 2151 | . . 3 |
4 | eqeq1 2146 | . . 3 | |
5 | moeq 2859 | . . . 4 | |
6 | 5 | a1i 9 | . . 3 |
7 | fvmptg.2 | . . . 4 | |
8 | df-mpt 3991 | . . . 4 | |
9 | 7, 8 | eqtri 2160 | . . 3 |
10 | 3, 4, 6, 9 | fvopab3ig 5495 | . 2 |
11 | 1, 10 | mpi 15 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wmo 2000 copab 3988 cmpt 3989 cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 |
This theorem is referenced by: fvmpt 5498 fvmpts 5499 fvmpt3 5500 fvmpt2 5504 f1mpt 5672 caofinvl 6004 1stvalg 6040 2ndvalg 6041 brtpos2 6148 rdgon 6283 frec0g 6294 freccllem 6299 frecfcllem 6301 frecsuclem 6303 sucinc 6341 sucinc2 6342 omcl 6357 oeicl 6358 oav2 6359 omv2 6361 fvdiagfn 6587 djulclr 6934 djurclr 6935 djulcl 6936 djurcl 6937 djulclb 6940 omp1eomlem 6979 ctmlemr 6993 nnnninf 7023 cardval3ex 7041 ceilqval 10079 frec2uzzd 10173 frec2uzsucd 10174 monoord2 10250 iseqf1olemqval 10260 iseqf1olemqk 10267 seq3f1olemqsum 10273 seq3f1oleml 10276 seq3f1o 10277 seq3distr 10286 ser3le 10291 hashinfom 10524 hashennn 10526 cjval 10617 reval 10621 imval 10622 cvg1nlemcau 10756 cvg1nlemres 10757 absval 10773 resqrexlemglsq 10794 resqrexlemga 10795 climmpt 11069 climle 11103 climcvg1nlem 11118 summodclem3 11149 summodclem2a 11150 zsumdc 11153 fsum3 11156 fsumcl2lem 11167 sumsnf 11178 isumadd 11200 fsumrev 11212 fsumshft 11213 fsummulc2 11217 iserabs 11244 isumlessdc 11265 divcnv 11266 trireciplem 11269 trirecip 11270 expcnvap0 11271 expcnvre 11272 expcnv 11273 explecnv 11274 geolim 11280 geolim2 11281 geo2lim 11285 geoisum 11286 geoisumr 11287 geoisum1 11288 geoisum1c 11289 cvgratz 11301 mertenslem2 11305 mertensabs 11306 eftvalcn 11363 efval 11367 efcvgfsum 11373 ege2le3 11377 efcj 11379 eftlub 11396 efgt1p2 11401 eflegeo 11408 sinval 11409 cosval 11410 tanvalap 11415 eirraplem 11483 phival 11889 crth 11900 phimullem 11901 ennnfonelemj0 11914 ennnfonelem0 11918 strnfvnd 11979 topnvalg 12132 toponsspwpwg 12189 tgval 12218 cldval 12268 ntrfval 12269 clsfval 12270 neifval 12309 neival 12312 ismet 12513 isxmet 12514 divcnap 12724 mulc1cncf 12745 djucllem 13007 nnsf 13199 peano3nninf 13201 nninfalllemn 13202 nninfself 13209 nninfsellemeqinf 13212 |
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