Theorem elrnmptg 5014

Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
rnmpt.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
elrnmptg	|- ( A. x e. A B e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)   F( x)   V( x)

Proof of Theorem elrnmptg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnmpt.1	. . . 4 |- F = ( x e. A |-> B )
2	1	rnmpt 5010	. . 3 |- ran F = { y | E. x e. A y = B }
3	2	eleq2i 2301	. 2 |- ( C e. ran F <-> C e. { y | E. x e. A y = B } )
4		r19.29 2682	. . . . 5 |- ( ( A. x e. A B e. V /\ E. x e. A C = B ) -> E. x e. A ( B e. V /\ C = B ) )
5		eleq1 2297	. . . . . . . 8 |- ( C = B -> ( C e. V <-> B e. V ) )
6	5	biimparc 299	. . . . . . 7 |- ( ( B e. V /\ C = B ) -> C e. V )
7		elex 2827	. . . . . . 7 |- ( C e. V -> C e. _V )
8	6, 7	syl 14	. . . . . 6 |- ( ( B e. V /\ C = B ) -> C e. _V )
9	8	rexlimivw 2658	. . . . 5 |- ( E. x e. A ( B e. V /\ C = B ) -> C e. _V )
10	4, 9	syl 14	. . . 4 |- ( ( A. x e. A B e. V /\ E. x e. A C = B ) -> C e. _V )
11	10	ex 115	. . 3 |- ( A. x e. A B e. V -> ( E. x e. A C = B -> C e. _V ) )
12		eqeq1 2241	. . . . 5 |- ( y = C -> ( y = B <-> C = B ) )
13	12	rexbidv 2545	. . . 4 |- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
14	13	elab3g 2971	. . 3 |- ( ( E. x e. A C = B -> C e. _V ) -> ( C e. { y | E. x e. A y = B } <-> E. x e. A C = B ) )
15	11, 14	syl 14	. 2 |- ( A. x e. A B e. V -> ( C e. { y | E. x e. A y = B } <-> E. x e. A C = B ) )
16	3, 15	bitrid 192	1 |- ( A. x e. A B e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   _Vcvv 2815   |-> cmpt 4176   ran crn 4755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-mpt 4178  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by:  elrnmpti  5015  fliftel  5972  2sqlem1  16113