Theorem elrnmptg 4915

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 4911	. . 3 |- ran F = { y | E. x e. A y = B }
3	2	eleq2i 2260	. 2 |- ( C e. ran F <-> C e. { y | E. x e. A y = B } )
4		r19.29 2631	. . . . 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 2256	. . . . . . . 8 |- ( C = B -> ( C e. V <-> B e. V ) )
6	5	biimparc 299	. . . . . . 7 |- ( ( B e. V /\ C = B ) -> C e. V )
7		elex 2771	. . . . . . 7 |- ( C e. V -> C e. _V )
8	6, 7	syl 14	. . . . . 6 |- ( ( B e. V /\ C = B ) -> C e. _V )
9	8	rexlimivw 2607	. . . . 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 2200	. . . . 5 |- ( y = C -> ( y = B <-> C = B ) )
13	12	rexbidv 2495	. . . 4 |- ( y = C -> ( E. x e. A y = B <-> E. x e. A C = B ) )
14	13	elab3g 2912	. . 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 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   |-> cmpt 4091   ran crn 4661

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-mpt 4093  df-cnv 4668  df-dm 4670  df-rn 4671

This theorem is referenced by:  elrnmpti  4916  fliftel  5837  2sqlem1  15271