Theorem fvresex 7139

Description: Existence of the class of values of a restricted class. (Contributed by NM, 14-Nov-1995.) (Revised by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
fvresex.1	|- A e. _V

Assertion

Ref	Expression
fvresex	|- { y | E. x y = ( ( F |` A ) ` x ) } e. _V

Distinct variable groups:   x, y, A   x, F, y

Proof of Theorem fvresex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssv 3625	. . . . . . . 8 |- A C_ _V
2		resmpt 5449	. . . . . . . 8 |- ( A C_ _V -> ( ( z e. _V |-> ( F ` z ) ) |` A ) = ( z e. A |-> ( F ` z ) ) )
3	1, 2	ax-mp 5	. . . . . . 7 |- ( ( z e. _V |-> ( F ` z ) ) |` A ) = ( z e. A |-> ( F ` z ) )
4	3	fveq1i 6192	. . . . . 6 |- ( ( ( z e. _V |-> ( F ` z ) ) |` A ) ` x ) = ( ( z e. A |-> ( F ` z ) ) ` x )
5		vex 3203	. . . . . . . 8 |- x e. _V
6		fveq2 6191	. . . . . . . . 9 |- ( z = x -> ( F ` z ) = ( F ` x ) )
7		eqid 2622	. . . . . . . . 9 |- ( z e. _V |-> ( F ` z ) ) = ( z e. _V |-> ( F ` z ) )
8		fvex 6201	. . . . . . . . 9 |- ( F ` x ) e. _V
9	6, 7, 8	fvmpt 6282	. . . . . . . 8 |- ( x e. _V -> ( ( z e. _V |-> ( F ` z ) ) ` x ) = ( F ` x ) )
10	5, 9	ax-mp 5	. . . . . . 7 |- ( ( z e. _V |-> ( F ` z ) ) ` x ) = ( F ` x )
11		fveqres 6230	. . . . . . 7 |- ( ( ( z e. _V |-> ( F ` z ) ) ` x ) = ( F ` x ) -> ( ( ( z e. _V |-> ( F ` z ) ) |` A ) ` x ) = ( ( F |` A ) ` x ) )
12	10, 11	ax-mp 5	. . . . . 6 |- ( ( ( z e. _V |-> ( F ` z ) ) |` A ) ` x ) = ( ( F |` A ) ` x )
13	4, 12	eqtr3i 2646	. . . . 5 |- ( ( z e. A |-> ( F ` z ) ) ` x ) = ( ( F |` A ) ` x )
14	13	eqeq2i 2634	. . . 4 |- ( y = ( ( z e. A |-> ( F ` z ) ) ` x ) <-> y = ( ( F |` A ) ` x ) )
15	14	exbii 1774	. . 3 |- ( E. x y = ( ( z e. A |-> ( F ` z ) ) ` x ) <-> E. x y = ( ( F |` A ) ` x ) )
16	15	abbii 2739	. 2 |- { y | E. x y = ( ( z e. A |-> ( F ` z ) ) ` x ) } = { y | E. x y = ( ( F |` A ) ` x ) }
17		fvresex.1	. . . 4 |- A e. _V
18	17	mptex 6486	. . 3 |- ( z e. A |-> ( F ` z ) ) e. _V
19	18	fvclex 7138	. 2 |- { y | E. x y = ( ( z e. A |-> ( F ` z ) ) ` x ) } e. _V
20	16, 19	eqeltrri 2698	1 |- { y | E. x y = ( ( F |` A ) ` x ) } e. _V

Syntax hints:   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   |` cres 5116   ` 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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by: (None)