Theorem eldmrexrn 5329

Description: For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)

Assertion

Ref	Expression
eldmrexrn	|- ( Fun F -> ( Y e. dom F -> E. x e. ran F x = ( F ` Y ) ) )

Distinct variable groups:   x, F   x, Y

Proof of Theorem eldmrexrn

Step	Hyp	Ref	Expression
1		fvelrn 5319	. . 3 |- ( ( Fun F /\ Y e. dom F ) -> ( F ` Y ) e. ran F )
2		eqid 2081	. . 3 |- ( F ` Y ) = ( F ` Y )
3		eqeq1 2087	. . . 4 |- ( x = ( F ` Y ) -> ( x = ( F ` Y ) <-> ( F ` Y ) = ( F ` Y ) ) )
4	3	rspcev 2701	. . 3 |- ( ( ( F ` Y ) e. ran F /\ ( F ` Y ) = ( F ` Y ) ) -> E. x e. ran F x = ( F ` Y ) )
5	1, 2, 4	sylancl 404	. 2 |- ( ( Fun F /\ Y e. dom F ) -> E. x e. ran F x = ( F ` Y ) )
6	5	ex 113	1 |- ( Fun F -> ( Y e. dom F -> E. x e. ran F x = ( F ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   E.wrex 2349   dom cdm 4363   ran crn 4364   Fun wfun 4916   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by: (None)