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Theorem elunirn 5318
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 3546 . 2  U. ran  F  ran  F
2 funfn 4845 . . . . . . . 8  Fun 
F  F  Fn  dom  F
3 fvelrnb 5134 . . . . . . . 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 2436 . . . . . 6  dom  F  F `  dom  F F `
75, 6syl6bbr 187 . . . . 5  Fun 
F  ran  F  dom  F  F `
8 eleq2 2074 . . . . . . 7  F `  F `
98biimparc 283 . . . . . 6  F `  F `
109reximi 2385 . . . . 5  dom  F  F `  dom  F  F `
117, 10syl6bi 152 . . . 4  Fun 
F  ran  F  dom  F  F `
1211exlimdv 1673 . . 3  Fun 
F  ran  F  dom  F  F `
13 fvelrn 5211 . . . . 5  Fun  F  dom  F  F `  ran  F
14 funfvex 5105 . . . . . 6  Fun  F  dom  F  F `  _V
15 eleq2 2074 . . . . . . . 8  F `  F `
16 eleq1 2073 . . . . . . . 8  F `  ran  F  F `

ran  F
1715, 16anbi12d 442 . . . . . . 7  F `  ran  F  F `  F `  ran  F
1817spcegv 2609 . . . . . 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 2402 . . 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 1223  wex 1354   wcel 1366  wrex 2276   _Vcvv 2526   U.cuni 3543   dom cdm 4260   ran crn 4261   Fun wfun 4811    Fn wfn 4812   ` cfv 4817
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-sbc 2733  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-iota 4782  df-fun 4819  df-fn 4820  df-fv 4825
This theorem is referenced by:  fnunirn  5319
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