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Theorem elrnrexdmb 5253
Description: For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Assertion
Ref Expression
elrnrexdmb  Fun 
F  Y  ran  F  dom  F  Y  F `
Distinct variable groups:   , F   , Y

Proof of Theorem elrnrexdmb
StepHypRef Expression
1 funfn 4877 . . 3  Fun 
F  F  Fn  dom  F
2 fvelrnb 5167 . . 3  F  Fn  dom  F  Y  ran  F  dom  F F `  Y
31, 2sylbi 114 . 2  Fun 
F  Y  ran  F  dom  F F `
 Y
4 eqcom 2042 . . 3  Y  F `  F `  Y
54rexbii 2328 . 2  dom  F  Y  F `  dom  F F `  Y
63, 5syl6bbr 187 1  Fun 
F  Y  ran  F  dom  F  Y  F `
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1243   wcel 1393  wrex 2304   dom cdm 4291   ran crn 4292   Fun wfun 4842    Fn wfn 4843   ` cfv 4848
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-uni 3575  df-br 3759  df-opab 3813  df-mpt 3814  df-id 4024  df-xp 4297  df-rel 4298  df-cnv 4299  df-co 4300  df-dm 4301  df-rn 4302  df-iota 4813  df-fun 4850  df-fn 4851  df-fv 4856
This theorem is referenced by: (None)
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