Theorem elrnmptg 4856

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 4852	. . 3 |- ran F = { y | E. x e. A y = B }
3	2	eleq2i 2233	. 2 |- ( C e. ran F <-> C e. { y | E. x e. A y = B } )
4		r19.29 2603	. . . . 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 2229	. . . . . . . 8 |- ( C = B -> ( C e. V <-> B e. V ) )
6	5	biimparc 297	. . . . . . 7 |- ( ( B e. V /\ C = B ) -> C e. V )
7		elex 2737	. . . . . . 7 |- ( C e. V -> C e. _V )
8	6, 7	syl 14	. . . . . 6 |- ( ( B e. V /\ C = B ) -> C e. _V )
9	8	rexlimivw 2579	. . . . 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 114	. . 3 |- ( A. x e. A B e. V -> ( E. x e. A C = B -> C e. _V ) )
12		eqeq1 2172	. . . . 5 |- ( y = C -> ( y = B <-> C = B ) )
13	12	rexbidv 2467	. . . 4 |- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
14	13	elab3g 2877	. . 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	syl5bb 191	1 |- ( A. x e. A B e. V -> ( C e. ran F <-> E. x e. A C = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   |-> cmpt 4043   ran crn 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-mpt 4045  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by:  elrnmpti  4857  fliftel  5761  2sqlem1  13590