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Mirrors > Home > ILE Home > Th. List > mptexg | Unicode version |
Description: If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
mptexg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 5156 | . 2 | |
2 | eqid 2137 | . . . 4 | |
3 | 2 | dmmptss 5030 | . . 3 |
4 | ssexg 4062 | . . 3 | |
5 | 3, 4 | mpan 420 | . 2 |
6 | funex 5636 | . 2 | |
7 | 1, 5, 6 | sylancr 410 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wcel 1480 cvv 2681 wss 3066 cmpt 3984 cdm 4534 wfun 5112 |
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 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 |
This theorem is referenced by: mptex 5639 offval 5982 abrexexg 6009 xpexgALT 6024 offval3 6025 iunon 6174 mptelixpg 6621 updjud 6960 mkvprop 7025 iseqf1olemqpcl 10262 seq3f1olemqsum 10266 seq3f1olemstep 10267 negfi 10992 climmpt 11062 restval 12115 ntrfval 12258 clsfval 12259 neifval 12298 cnprcl2k 12364 upxp 12430 |
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