Theorem elrnrexdm 5786

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 2232	. . . . . 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 2241	. . . . 5 |- ( y = Y -> ( Y = y <-> Y = Y ) )
5	4	rspcev 2910	. . . 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 5356	. . 3 |- ( Fun F <-> F Fn dom F )
9		eqeq2 2241	. . . 4 |- ( y = ( F ` x ) -> ( Y = y <-> Y = ( F ` x ) ) )
10	9	rexrn 5784	. . 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 1397   e. wcel 2202   E.wrex 2511   dom cdm 4725   ran crn 4726   Fun wfun 5320   Fn wfn 5321   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  cc2lem  7484  ennnfonelemrnh  13036  ennnfonelemf1  13038  upgredg  15994  exmidsbthrlem  16626