Theorem eldmrexrn 5778

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 5768	. . 3 |- ( ( Fun F /\ Y e. dom F ) -> ( F ` Y ) e. ran F )
2		eqid 2229	. . 3 |- ( F ` Y ) = ( F ` Y )
3		eqeq1 2236	. . . 4 |- ( x = ( F ` Y ) -> ( x = ( F ` Y ) <-> ( F ` Y ) = ( F ` Y ) ) )
4	3	rspcev 2907	. . 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 1395   e. wcel 2200   E.wrex 2509   dom cdm 4719   ran crn 4720   Fun wfun 5312   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326

This theorem is referenced by: (None)