Theorem fvn0fvelrn 6430

Description: If the value of a function is not null, the value is an element of the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

Assertion

Ref	Expression
fvn0fvelrn	|- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F )

Proof of Theorem fvn0fvelrn

Step	Hyp	Ref	Expression
1		fvfundmfvn0 6226	. 2 |- ( ( F ` X ) =/= (/) -> ( X e. dom F /\ Fun ( F |` { X } ) ) )
2		eldmressnsn 5439	. . . 4 |- ( X e. dom F -> X e. dom ( F |` { X } ) )
3		fvelrn 6352	. . . . . . 7 |- ( ( Fun ( F |` { X } ) /\ X e. dom ( F |` { X } ) ) -> ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) )
4		pm3.2 463	. . . . . . 7 |- ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) -> ( X e. dom F -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) /\ X e. dom F ) ) )
5	3, 4	syl 17	. . . . . 6 |- ( ( Fun ( F |` { X } ) /\ X e. dom ( F |` { X } ) ) -> ( X e. dom F -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) /\ X e. dom F ) ) )
6	5	ex 450	. . . . 5 |- ( Fun ( F |` { X } ) -> ( X e. dom ( F |` { X } ) -> ( X e. dom F -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) /\ X e. dom F ) ) ) )
7	6	com13 88	. . . 4 |- ( X e. dom F -> ( X e. dom ( F |` { X } ) -> ( Fun ( F |` { X } ) -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) /\ X e. dom F ) ) ) )
8	2, 7	mpd 15	. . 3 |- ( X e. dom F -> ( Fun ( F |` { X } ) -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) /\ X e. dom F ) ) )
9	8	imp 445	. 2 |- ( ( X e. dom F /\ Fun ( F |` { X } ) ) -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) /\ X e. dom F ) )
10		fvressn 6429	. . . . 5 |- ( X e. dom F -> ( ( F |` { X } ) ` X ) = ( F ` X ) )
11	10	eleq1d 2686	. . . 4 |- ( X e. dom F -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) <-> ( F ` X ) e. ran ( F |` { X } ) ) )
12		fvrnressn 6428	. . . 4 |- ( X e. dom F -> ( ( F ` X ) e. ran ( F |` { X } ) -> ( F ` X ) e. ran F ) )
13	11, 12	sylbid 230	. . 3 |- ( X e. dom F -> ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) -> ( F ` X ) e. ran F ) )
14	13	impcom 446	. 2 |- ( ( ( ( F |` { X } ) ` X ) e. ran ( F |` { X } ) /\ X e. dom F ) -> ( F ` X ) e. ran F )
15	1, 9, 14	3syl 18	1 |- ( ( F ` X ) =/= (/) -> ( F ` X ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   =/= wne 2794   (/)c0 3915   {csn 4177   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  wlkvtxiedg  26520