Theorem eldmrexrn 5823

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 5813	. . 3 |- ( ( Fun F /\ Y e. dom F ) -> ( F ` Y ) e. ran F )
2		eqid 2234	. . 3 |- ( F ` Y ) = ( F ` Y )
3		eqeq1 2241	. . . 4 |- ( x = ( F ` Y ) -> ( x = ( F ` Y ) <-> ( F ` Y ) = ( F ` Y ) ) )
4	3	rspcev 2923	. . 3 |- ( ( ( F ` Y ) e. ran F /\ ( F ` Y ) = ( F ` Y ) ) -> E. x e. ran F x = ( F ` Y ) )
5	1, 2, 4	sylancl 413	. 2 |- ( ( Fun F /\ Y e. dom F ) -> E. x e. ran F x = ( F ` Y ) )
6	5	ex 115	1 |- ( Fun F -> ( Y e. dom F -> E. x e. ran F x = ( F ` Y ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   E.wrex 2523   dom cdm 4754   ran crn 4755   Fun wfun 5351   ` cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by: (None)