Theorem elrnmptg 5009

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 5005	. . 3 |- ran F = { y | E. x e. A y = B }
3	2	eleq2i 2299	. 2 |- ( C e. ran F <-> C e. { y | E. x e. A y = B } )
4		r19.29 2680	. . . . 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 2295	. . . . . . . 8 |- ( C = B -> ( C e. V <-> B e. V ) )
6	5	biimparc 299	. . . . . . 7 |- ( ( B e. V /\ C = B ) -> C e. V )
7		elex 2825	. . . . . . 7 |- ( C e. V -> C e. _V )
8	6, 7	syl 14	. . . . . 6 |- ( ( B e. V /\ C = B ) -> C e. _V )
9	8	rexlimivw 2656	. . . . 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 2239	. . . . 5 |- ( y = C -> ( y = B <-> C = B ) )
13	12	rexbidv 2543	. . . 4 |- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
14	13	elab3g 2968	. . 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 2203   {cab 2218   A.wral 2520   E.wrex 2521   _Vcvv 2813   |-> cmpt 4171   ran crn 4750

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-mpt 4173  df-cnv 4757  df-dm 4759  df-rn 4760

This theorem is referenced by:  elrnmpti  5010  fliftel  5966  2sqlem1  15987