Theorem elrnmptg 4918

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 4914	. . 3 |- ran F = { y | E. x e. A y = B }
3	2	eleq2i 2263	. 2 |- ( C e. ran F <-> C e. { y | E. x e. A y = B } )
4		r19.29 2634	. . . . 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 2259	. . . . . . . 8 |- ( C = B -> ( C e. V <-> B e. V ) )
6	5	biimparc 299	. . . . . . 7 |- ( ( B e. V /\ C = B ) -> C e. V )
7		elex 2774	. . . . . . 7 |- ( C e. V -> C e. _V )
8	6, 7	syl 14	. . . . . 6 |- ( ( B e. V /\ C = B ) -> C e. _V )
9	8	rexlimivw 2610	. . . . 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 2203	. . . . 5 |- ( y = C -> ( y = B <-> C = B ) )
13	12	rexbidv 2498	. . . 4 |- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
14	13	elab3g 2915	. . 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 1364   e. wcel 2167   {cab 2182   A.wral 2475   E.wrex 2476   _Vcvv 2763   |-> cmpt 4094   ran crn 4664

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095  df-mpt 4096  df-cnv 4671  df-dm 4673  df-rn 4674

This theorem is referenced by:  elrnmpti  4919  fliftel  5840  2sqlem1  15355