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Mirrors > Home > MPE Home > Th. List > rnex | Structured version Visualization version GIF version |
Description: The range of a set is a set. Corollary 6.8(3) of [TakeutiZaring] p. 26. Similar to Lemma 3D of [Enderton] p. 41. (Contributed by NM, 7-Jul-2008.) |
Ref | Expression |
---|---|
dmex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
rnex | ⊢ ran 𝐴 ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmex.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | rnexg 6990 | . 2 ⊢ (𝐴 ∈ V → ran 𝐴 ∈ V) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ran 𝐴 ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 ran crn 5039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: elxp4 7003 elxp5 7004 ffoss 7020 fvclex 7031 abrexex 7033 wemoiso2 7045 2ndval 7062 fo2nd 7080 ixpsnf1o 7834 bren 7850 mapen 8009 ssenen 8019 sucdom2 8041 fodomfib 8125 hartogslem1 8330 brwdom 8355 unxpwdom2 8376 noinfep 8440 r0weon 8718 fseqen 8733 acnlem 8754 infpwfien 8768 aceq3lem 8826 dfac4 8828 dfac5 8834 dfac2 8836 dfac9 8841 dfac12lem2 8849 dfac12lem3 8850 infmap2 8923 cfflb 8964 infpssr 9013 fin23lem14 9038 fin23lem16 9040 fin23lem17 9043 fin23lem38 9054 fin23lem39 9055 axdc2lem 9153 axdc3lem2 9156 axcclem 9162 ttukeylem6 9219 wunex2 9439 wuncval2 9448 intgru 9515 wfgru 9517 qexALT 11679 hashfacen 13095 shftfval 13658 vdwapval 15515 restfn 15908 prdsval 15938 wunfunc 16382 wunnat 16439 arwval 16516 catcfuccl 16582 catcxpccl 16670 yon11 16727 yon12 16728 yon2 16729 yonpropd 16731 oppcyon 16732 yonffth 16747 yoniso 16748 plusffval 17070 sylow1lem2 17837 sylow2blem1 17858 sylow2blem2 17859 sylow3lem1 17865 sylow3lem6 17870 dmdprd 18220 dprdval 18225 staffval 18670 scaffval 18704 lpival 19066 ipffval 19812 cmpsub 21013 2ndcsep 21072 1stckgen 21167 kgencn2 21170 txcmplem1 21254 blbas 22045 met1stc 22136 psmetutop 22182 nmfval 22203 qtopbaslem 22372 dchrptlem2 24790 dchrptlem3 24791 ishpg 25451 edgaval 25794 edgval 25868 bafval 26843 vsfval 26872 foresf1o 28727 locfinreflem 29235 cmpcref 29245 ordtconlem1 29298 qqhval 29346 sigapildsys 29552 dya2icoseg2 29667 dya2iocuni 29672 sxbrsigalem2 29675 sxbrsigalem5 29677 omssubadd 29689 mvtval 30651 mvrsval 30656 mstaval 30695 trpredex 30981 brrestrict 31226 relowlssretop 32387 lindsdom 32573 indexdom 32699 heiborlem1 32780 isdrngo2 32927 isrngohom 32934 idlval 32982 isidl 32983 igenval 33030 lsatset 33295 dicval 35483 trclexi 36946 rtrclexi 36947 dfrtrcl5 36955 dfrcl2 36985 stoweidlem59 38952 fourierdlem71 39070 salgensscntex 39238 aacllem 42356 |
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