Theorem elrnrexdm 5567

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 2141	. . . . . 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 2150	. . . . 5 |- ( y = Y -> ( Y = y <-> Y = Y ) )
5	4	rspcev 2793	. . . 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 5161	. . 3 |- ( Fun F <-> F Fn dom F )
9		eqeq2 2150	. . . 4 |- ( y = ( F ` x ) -> ( Y = y <-> Y = ( F ` x ) ) )
10	9	rexrn 5565	. . 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 1332   e. wcel 1481   E.wrex 2418   dom cdm 4547   ran crn 4548   Fun wfun 5125   Fn wfn 5126   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139

This theorem is referenced by:  cc2lem  7098  ennnfonelemrnh  11965  ennnfonelemf1  11967  exmidsbthrlem  13392