Theorem afvelrn 41248

Description: A function's value belongs to its range, analogous to fvelrn 6352. (Contributed by Alexander van der Vekens, 25-May-2017.)

Assertion

Ref	Expression
afvelrn	|- ( ( Fun F /\ A e. dom F ) -> ( F''' A ) e. ran F )

Proof of Theorem afvelrn

Step	Hyp	Ref	Expression
1		funres 5929	. . . . . 6 |- ( Fun F -> Fun ( F |` { A } ) )
2	1	anim1i 592	. . . . 5 |- ( ( Fun F /\ A e. dom F ) -> ( Fun ( F |` { A } ) /\ A e. dom F ) )
3	2	ancomd 467	. . . 4 |- ( ( Fun F /\ A e. dom F ) -> ( A e. dom F /\ Fun ( F |` { A } ) ) )
4		df-dfat 41196	. . . 4 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
5	3, 4	sylibr 224	. . 3 |- ( ( Fun F /\ A e. dom F ) -> F defAt A )
6		afvfundmfveq 41218	. . . 4 |- ( F defAt A -> ( F''' A ) = ( F ` A ) )
7	6	eqcomd 2628	. . 3 |- ( F defAt A -> ( F ` A ) = ( F''' A ) )
8	5, 7	syl 17	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) = ( F''' A ) )
9		fvelrn 6352	. 2 |- ( ( Fun F /\ A e. dom F ) -> ( F ` A ) e. ran F )
10	8, 9	eqeltrrd 2702	1 |- ( ( Fun F /\ A e. dom F ) -> ( F''' A ) e. ran F )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   ` cfv 5888   defAt wdfat 41193  '''cafv 41194

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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-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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-dfat 41196  df-afv 41197

This theorem is referenced by:  fnafvelrn  41249