Theorem elrnrexdm 5606

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, 8-Dec-2017.)

Assertion

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

Distinct variable groups:   x, F   x, Y

Proof of Theorem elrnrexdm

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2158	. . . . . 6 |- ( Y e. ran F -> Y = Y )
2	1	ancli 321	. . . . 5 |- ( Y e. ran F -> ( Y e. ran F /\ Y = Y ) )
3	2	adantl 275	. . . 4 |- ( ( Fun F /\ Y e. ran F ) -> ( Y e. ran F /\ Y = Y ) )
4		eqeq2 2167	. . . . 5 |- ( y = Y -> ( Y = y <-> Y = Y ) )
5	4	rspcev 2816	. . . 4 |- ( ( Y e. ran F /\ Y = Y ) -> E. y e. ran F Y = y )
6	3, 5	syl 14	. . 3 |- ( ( Fun F /\ Y e. ran F ) -> E. y e. ran F Y = y )
7	6	ex 114	. 2 |- ( Fun F -> ( Y e. ran F -> E. y e. ran F Y = y ) )
8		funfn 5200	. . 3 |- ( Fun F <-> F Fn dom F )
9		eqeq2 2167	. . . 4 |- ( y = ( F ` x ) -> ( Y = y <-> Y = ( F ` x ) ) )
10	9	rexrn 5604	. . 3 |- ( F Fn dom F -> ( E. y e. ran F Y = y <-> E. x e. dom F Y = ( F ` x ) ) )
11	8, 10	sylbi 120	. 2 |- ( Fun F -> ( E. y e. ran F Y = y <-> E. x e. dom F Y = ( F ` x ) ) )
12	7, 11	sylibd 148	1 |- ( Fun F -> ( Y e. ran F -> E. x e. dom F Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   E.wrex 2436   dom cdm 4586   ran crn 4587   Fun wfun 5164   Fn wfn 5165   ` cfv 5170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-fv 5178

This theorem is referenced by:  cc2lem  7186  ennnfonelemrnh  12145  ennnfonelemf1  12147  exmidsbthrlem  13593