Theorem elrnrexdm 5697

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 2194	. . . . . 6 |- ( Y e. ran F -> Y = Y )
2	1	ancli 323	. . . . 5 |- ( Y e. ran F -> ( Y e. ran F /\ Y = Y ) )
3	2	adantl 277	. . . 4 |- ( ( Fun F /\ Y e. ran F ) -> ( Y e. ran F /\ Y = Y ) )
4		eqeq2 2203	. . . . 5 |- ( y = Y -> ( Y = y <-> Y = Y ) )
5	4	rspcev 2864	. . . 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 115	. 2 |- ( Fun F -> ( Y e. ran F -> E. y e. ran F Y = y ) )
8		funfn 5284	. . 3 |- ( Fun F <-> F Fn dom F )
9		eqeq2 2203	. . . 4 |- ( y = ( F ` x ) -> ( Y = y <-> Y = ( F ` x ) ) )
10	9	rexrn 5695	. . 3 |- ( F Fn dom F -> ( E. y e. ran F Y = y <-> E. x e. dom F Y = ( F ` x ) ) )
11	8, 10	sylbi 121	. 2 |- ( Fun F -> ( E. y e. ran F Y = y <-> E. x e. dom F Y = ( F ` x ) ) )
12	7, 11	sylibd 149	1 |- ( Fun F -> ( Y e. ran F -> E. x e. dom F Y = ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   E.wrex 2473   dom cdm 4659   ran crn 4660   Fun wfun 5248   Fn wfn 5249   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262

This theorem is referenced by:  cc2lem  7326  ennnfonelemrnh  12573  ennnfonelemf1  12575  exmidsbthrlem  15512