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Mirrors > Home > MPE Home > Th. List > rnexg | 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, 31-Mar-1995.) |
Ref | Expression |
---|---|
rnexg | ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 7456 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
2 | uniexg 7456 | . 2 ⊢ (∪ 𝐴 ∈ V → ∪ ∪ 𝐴 ∈ V) | |
3 | ssun2 4146 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
4 | dmrnssfld 5834 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
5 | 3, 4 | sstri 3973 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
6 | ssexg 5218 | . . 3 ⊢ ((ran 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ∈ V) → ran 𝐴 ∈ V) | |
7 | 5, 6 | mpan 686 | . 2 ⊢ (∪ ∪ 𝐴 ∈ V → ran 𝐴 ∈ V) |
8 | 1, 2, 7 | 3syl 18 | 1 ⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 Vcvv 3492 ∪ cun 3931 ⊆ wss 3933 ∪ cuni 4830 dom cdm 5548 ran crn 5549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: rnex 7606 imaexg 7609 xpexr 7612 xpexr2 7613 soex 7615 cnvexg 7618 coexg 7623 cofunexg 7639 funrnex 7644 abrexexg 7651 tposexg 7895 iunon 7965 onoviun 7969 tz7.44lem1 8030 tz7.44-3 8033 fopwdom 8613 disjen 8662 domss2 8664 domssex 8666 hartogslem2 8995 djuexb 9326 dfac12lem2 9558 unirnfdomd 9977 focdmex 13699 hashimarn 13789 trclexlem 14342 relexp0g 14369 relexpsucnnr 14372 restval 16688 prdsbas 16718 prdsplusg 16719 prdsmulr 16720 prdsvsca 16721 prdshom 16728 sscpwex 17073 sylow1lem4 18655 sylow3lem2 18682 sylow3lem3 18683 lsmvalx 18693 txindislem 22169 xkoptsub 22190 fmfnfmlem3 22492 fmfnfmlem4 22493 ustuqtoplem 22775 ustuqtop0 22776 utopsnneiplem 22783 efabl 25061 efsubm 25062 perpln1 26423 perpln2 26424 isperp 26425 lmif 26498 islmib 26500 isgrpo 28201 grpoinvfval 28226 grpodivfval 28238 isvcOLD 28283 isnv 28316 abrexexd 30196 acunirnmpt 30332 acunirnmpt2 30333 acunirnmpt2f 30334 fnpreimac 30344 locfinreflem 31003 esumrnmpt2 31226 sxsigon 31350 omssubadd 31457 carsgclctunlem2 31476 pmeasadd 31482 sitgclg 31499 bnj1366 32000 ptrest 34772 elghomlem1OLD 35044 elghomlem2OLD 35045 isrngod 35057 iscringd 35157 imaexALTV 35468 xrnresex 35534 dfcnvrefrels2 35646 dfcnvrefrels3 35647 lmhmlnmsplit 39565 rclexi 39853 rtrclexlem 39854 trclubgNEW 39856 cnvrcl0 39863 dfrtrcl5 39867 relexpmulg 39933 relexp01min 39936 relexpxpmin 39940 unirnmap 41347 unirnmapsn 41353 ssmapsn 41355 fourierdlem70 42338 fourierdlem71 42339 fourierdlem80 42348 meadjiunlem 42624 omeiunle 42676 fexafv2ex 43296 |
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