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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 5254 |
. 2
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2 | eqid 2177 |
. . . 4
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3 | 2 | dmmptss 5125 |
. . 3
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4 | ssexg 4142 |
. . 3
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5 | 3, 4 | mpan 424 |
. 2
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6 | funex 5739 |
. 2
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7 | 1, 5, 6 | sylancr 414 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 |
This theorem is referenced by: mptex 5742 mptexd 5743 offval 6089 abrexexg 6118 xpexgALT 6133 offval3 6134 iunon 6284 mptelixpg 6733 updjud 7080 mkvprop 7155 cc3 7266 iseqf1olemqpcl 10495 seq3f1olemqsum 10499 seq3f1olemstep 10500 negfi 11235 climmpt 11307 restval 12693 ntrfval 13570 clsfval 13571 neifval 13610 cnprcl2k 13676 upxp 13742 |
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