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Mirrors > Home > MPE Home > Th. List > mpt2ex | Structured version Visualization version Unicode version |
Description: If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Mario Carneiro, 20-Dec-2013.) |
Ref | Expression |
---|---|
mpt2ex.1 | |
mpt2ex.2 |
Ref | Expression |
---|---|
mpt2ex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2ex.1 | . 2 | |
2 | mpt2ex.2 | . . 3 | |
3 | 2 | rgenw 2924 | . 2 |
4 | eqid 2622 | . . 3 | |
5 | 4 | mpt2exxg 7244 | . 2 |
6 | 1, 3, 5 | mp2an 708 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wcel 1990 wral 2912 cvv 3200 cmpt2 6652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 |
This theorem is referenced by: qexALT 11803 ruclem13 14971 vdwapfval 15675 prdsco 16128 imasvsca 16180 homffval 16350 comfffval 16358 comffval 16359 comfffn 16364 comfeq 16366 oppccofval 16376 monfval 16392 sectffval 16410 invffval 16418 cofu1st 16543 cofu2nd 16545 cofucl 16548 natfval 16606 fuccofval 16619 fucco 16622 coafval 16714 setcco 16733 catchomfval 16748 catccofval 16750 catcco 16751 estrcco 16770 xpcval 16817 xpchomfval 16819 xpccofval 16822 xpcco 16823 1stf1 16832 1stf2 16833 2ndf1 16835 2ndf2 16836 1stfcl 16837 2ndfcl 16838 prf1 16840 prf2fval 16841 prfcl 16843 prf1st 16844 prf2nd 16845 evlf2 16858 evlf1 16860 evlfcl 16862 curf1fval 16864 curf11 16866 curf12 16867 curf1cl 16868 curf2 16869 curfcl 16872 hof1fval 16893 hof2fval 16895 hofcl 16899 yonedalem3 16920 mgmnsgrpex 17418 sgrpnmndex 17419 grpsubfval 17464 mulgfval 17542 symgplusg 17809 lsmfval 18053 pj1fval 18107 dvrfval 18684 psrmulr 19384 psrvscafval 19390 evlslem2 19512 mamufval 20191 mvmulfval 20348 isphtpy 22780 pcofval 22810 q1pval 23913 r1pval 23916 motplusg 25437 midf 25668 ismidb 25670 ttgval 25755 ebtwntg 25862 ecgrtg 25863 elntg 25864 wwlksnon 26738 wspthsnon 26739 vsfval 27488 dipfval 27557 smatfval 29861 lmatval 29879 qqhval 30018 dya2iocuni 30345 sxbrsigalem5 30350 sitmval 30411 signswplusg 30632 reprval 30688 mclsrcl 31458 mclsval 31460 ldualfvs 34423 paddfval 35083 tgrpopr 36035 erngfplus 36090 erngfmul 36093 erngfplus-rN 36098 erngfmul-rN 36101 dvafvadd 36302 dvafvsca 36304 dvaabl 36313 dvhfvadd 36380 dvhfvsca 36389 djafvalN 36423 djhfval 36686 hlhilip 37240 mendplusgfval 37755 mendmulrfval 37757 mendvscafval 37760 hoidmvval 40791 cznrng 41955 cznnring 41956 rngchomfvalALTV 41984 rngccofvalALTV 41987 rngccoALTV 41988 ringchomfvalALTV 42047 ringccofvalALTV 42050 ringccoALTV 42051 |
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