Theorem elrnmptg 5375

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 5371	. . 3 |- ran F = { y | E. x e. A y = B }
3	2	eleq2i 2693	. 2 |- ( C e. ran F <-> C e. { y | E. x e. A y = B } )
4		r19.29 3072	. . . . 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 2689	. . . . . . . 8 |- ( C = B -> ( C e. V <-> B e. V ) )
6	5	biimparc 504	. . . . . . 7 |- ( ( B e. V /\ C = B ) -> C e. V )
7	6	elexd 3214	. . . . . 6 |- ( ( B e. V /\ C = B ) -> C e. _V )
8	7	rexlimivw 3029	. . . . 5 |- ( E. x e. A ( B e. V /\ C = B ) -> C e. _V )
9	4, 8	syl 17	. . . 4 |- ( ( A. x e. A B e. V /\ E. x e. A C = B ) -> C e. _V )
10	9	ex 450	. . 3 |- ( A. x e. A B e. V -> ( E. x e. A C = B -> C e. _V ) )
11		eqeq1 2626	. . . . 5 |- ( y = C -> ( y = B <-> C = B ) )
12	11	rexbidv 3052	. . . 4 |- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
13	12	elab3g 3357	. . 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 ) )
14	10, 13	syl 17	. 2 |- ( A. x e. A B e. V -> ( C e. { y | E. x e. A y = B } <-> E. x e. A C = B ) )
15	3, 14	syl5bb 272	1 |- ( A. x e. A B e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   E.wrex 2913   _Vcvv 3200   |-> cmpt 4729   ran crn 5115

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-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-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  elrnmpti  5376