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Theorem elunirn 5348
Description: Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
Assertion
Ref Expression
elunirn  Fun 
F  U. ran  F  dom  F  F `
Distinct variable groups:   ,   , F

Proof of Theorem elunirn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eluni 3574 . 2  U. ran  F  ran  F
2 funfn 4874 . . . . . . . 8  Fun 
F  F  Fn  dom  F
3 fvelrnb 5164 . . . . . . . 8  F  Fn  dom  F  ran  F  dom  F F `
42, 3sylbi 114 . . . . . . 7  Fun 
F  ran  F  dom  F F `
54anbi2d 437 . . . . . 6  Fun 
F  ran  F  dom  F F `
6 r19.42v 2461 . . . . . 6  dom  F  F `  dom  F F `
75, 6syl6bbr 187 . . . . 5  Fun 
F  ran  F  dom  F  F `
8 eleq2 2098 . . . . . . 7  F `  F `
98biimparc 283 . . . . . 6  F `  F `
109reximi 2410 . . . . 5  dom  F  F `  dom  F  F `
117, 10syl6bi 152 . . . 4  Fun 
F  ran  F  dom  F  F `
1211exlimdv 1697 . . 3  Fun 
F  ran  F  dom  F  F `
13 fvelrn 5241 . . . . 5  Fun  F  dom  F  F `  ran  F
14 funfvex 5135 . . . . . 6  Fun  F  dom  F  F `  _V
15 eleq2 2098 . . . . . . . 8  F `  F `
16 eleq1 2097 . . . . . . . 8  F `  ran  F  F `

ran  F
1715, 16anbi12d 442 . . . . . . 7  F `  ran  F  F `  F `  ran  F
1817spcegv 2635 . . . . . 6  F `  _V  F `  F `  ran  F  ran  F
1914, 18syl 14 . . . . 5  Fun  F  dom  F  F `  F `  ran  F  ran  F
2013, 19mpan2d 404 . . . 4  Fun  F  dom  F  F `  ran  F
2120rexlimdva 2427 . . 3  Fun 
F  dom  F  F `  ran  F
2212, 21impbid 120 . 2  Fun 
F  ran  F 
dom  F  F `
231, 22syl5bb 181 1  Fun 
F  U. ran  F  dom  F  F `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390  wrex 2301   _Vcvv 2551   U.cuni 3571   dom cdm 4288   ran crn 4289   Fun wfun 4839    Fn wfn 4840   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fnunirn  5349
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