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| Mirrors > Home > MPE Home > Th. List > dmexg | Structured version Visualization version GIF version | ||
| Description: The domain of a set is a set. Corollary 6.8(2) of [TakeutiZaring] p. 26. (Contributed by NM, 7-Apr-1995.) |
| Ref | Expression |
|---|---|
| dmexg | ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uniexg 7760 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V) | |
| 2 | uniexg 7760 | . 2 ⊢ (∪ 𝐴 ∈ V → ∪ ∪ 𝐴 ∈ V) | |
| 3 | ssun1 4178 | . . . 4 ⊢ dom 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
| 4 | dmrnssfld 5984 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
| 5 | 3, 4 | sstri 3993 | . . 3 ⊢ dom 𝐴 ⊆ ∪ ∪ 𝐴 |
| 6 | ssexg 5323 | . . 3 ⊢ ((dom 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ∈ V) → dom 𝐴 ∈ V) | |
| 7 | 5, 6 | mpan 690 | . 2 ⊢ (∪ ∪ 𝐴 ∈ V → dom 𝐴 ∈ V) |
| 8 | 1, 2, 7 | 3syl 18 | 1 ⊢ (𝐴 ∈ 𝑉 → dom 𝐴 ∈ V) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ⊆ wss 3951 ∪ cuni 4907 dom cdm 5685 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: dmexd 7925 dmfex 7927 dmex 7931 iprc 7933 exse2 7939 xpexr2 7941 xpexcnv 7942 soex 7943 cnvexg 7946 coexg 7951 cofunexg 7973 offval3 8007 opabn1stprc 8083 suppval 8187 funsssuppss 8215 suppssov1 8222 suppssov2 8223 suppssfv 8227 tposexg 8265 tfrlem12 8429 tfrlem13 8430 erexb 8770 f1vrnfibi 9382 oion 9576 ttrclexg 9763 fpwwe2lem3 10673 hashfn 14414 hashfundm 14481 hashf1dmrn 14482 fundmge2nop0 14541 fun2dmnop0 14543 trclexlem 15033 relexp0g 15061 relexpsucnnr 15064 o1of2 15649 isofn 17819 ssclem 17863 ssc2 17866 ssctr 17869 subsubc 17898 resf1st 17939 resf2nd 17940 funcres 17941 dprddomprc 20020 dprdval0prc 20022 subgdmdprd 20054 dprd2da 20062 decpmatval0 22770 pmatcollpw3lem 22789 ordtbaslem 23196 ordtuni 23198 ordtbas2 23199 ordtbas 23200 ordttopon 23201 ordtopn1 23202 ordtopn2 23203 txindislem 23641 ordthmeolem 23809 ptcmplem2 24061 tuslem 24275 tuslemOLD 24276 dvnff 25959 bdayval 27693 noextend 27711 bdayfo 27722 vtxdgf 29489 fdifsuppconst 32698 ressupprn 32699 ofcfval3 34103 braew 34243 omsval 34295 sibfof 34342 sitmcl 34353 cndprobval 34435 tailf 36376 tailfb 36378 ismgmOLD 37857 dfcnvrefrels2 38529 dfcnvrefrels3 38530 rclexi 43628 rtrclexlem 43629 cnvrcl0 43638 dfrtrcl5 43642 relexpmulg 43723 relexp01min 43726 relexpxpmin 43730 unidmex 45055 caragenval 46508 caragenunidm 46523 itcoval0 48583 itcoval1 48584 |
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