Theorem fvrnressn 6428

Description: If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

Assertion

Ref	Expression
fvrnressn	|- ( X e. V -> ( ( F ` X ) e. ran ( F |` { X } ) -> ( F ` X ) e. ran F ) )

Proof of Theorem fvrnressn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5127	. . 3 |- ( F " { X } ) = ran ( F |` { X } )
2	1	eleq2i 2693	. 2 |- ( ( F ` X ) e. ( F " { X } ) <-> ( F ` X ) e. ran ( F |` { X } ) )
3		opeq1 4402	. . . . 5 |- ( x = X -> <. x , ( F ` X ) >. = <. X , ( F ` X ) >. )
4	3	eleq1d 2686	. . . 4 |- ( x = X -> ( <. x , ( F ` X ) >. e. F <-> <. X , ( F ` X ) >. e. F ) )
5	4	spcegv 3294	. . 3 |- ( X e. V -> ( <. X , ( F ` X ) >. e. F -> E. x <. x , ( F ` X ) >. e. F ) )
6		fvex 6201	. . . 4 |- ( F ` X ) e. _V
7		elimasng 5491	. . . 4 |- ( ( X e. V /\ ( F ` X ) e. _V ) -> ( ( F ` X ) e. ( F " { X } ) <-> <. X , ( F ` X ) >. e. F ) )
8	6, 7	mpan2 707	. . 3 |- ( X e. V -> ( ( F ` X ) e. ( F " { X } ) <-> <. X , ( F ` X ) >. e. F ) )
9		elrn2g 5313	. . . 4 |- ( ( F ` X ) e. _V -> ( ( F ` X ) e. ran F <-> E. x <. x , ( F ` X ) >. e. F ) )
10	6, 9	mp1i 13	. . 3 |- ( X e. V -> ( ( F ` X ) e. ran F <-> E. x <. x , ( F ` X ) >. e. F ) )
11	5, 8, 10	3imtr4d 283	. 2 |- ( X e. V -> ( ( F ` X ) e. ( F " { X } ) -> ( F ` X ) e. ran F ) )
12	2, 11	syl5bir 233	1 |- ( X e. V -> ( ( F ` X ) e. ran ( F |` { X } ) -> ( F ` X ) e. ran F ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   ran crn 5115   |` cres 5116   "cima 5117   ` 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-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-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fv 5896

This theorem is referenced by:  fvn0fvelrn  6430