Theorem ovelrn 5793

Description: A member of an operation's range is a value of the operation. (Contributed by NM, 7-Feb-2007.) (Revised by Mario Carneiro, 30-Jan-2014.)

Assertion

Ref	Expression
ovelrn	|- ( F Fn ( A X. B ) -> ( C e. ran F <-> E. x e. A E. y e. B C = ( x F y ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, F, y

Proof of Theorem ovelrn

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnrnov 5790	. . 3 |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )
2	1	eleq2d 2157	. 2 |- ( F Fn ( A X. B ) -> ( C e. ran F <-> C e. { z | E. x e. A E. y e. B z = ( x F y ) } ) )
3		elex 2630	. . . 4 |- ( C e. { z | E. x e. A E. y e. B z = ( x F y ) } -> C e. _V )
4	3	a1i 9	. . 3 |- ( F Fn ( A X. B ) -> ( C e. { z | E. x e. A E. y e. B z = ( x F y ) } -> C e. _V ) )
5		fnovex 5682	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( x F y ) e. _V )
6		eleq1 2150	. . . . . 6 |- ( C = ( x F y ) -> ( C e. _V <-> ( x F y ) e. _V ) )
7	5, 6	syl5ibrcom 155	. . . . 5 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( C = ( x F y ) -> C e. _V ) )
8	7	3expb 1144	. . . 4 |- ( ( F Fn ( A X. B ) /\ ( x e. A /\ y e. B ) ) -> ( C = ( x F y ) -> C e. _V ) )
9	8	rexlimdvva 2496	. . 3 |- ( F Fn ( A X. B ) -> ( E. x e. A E. y e. B C = ( x F y ) -> C e. _V ) )
10		eqeq1 2094	. . . . . 6 |- ( z = C -> ( z = ( x F y ) <-> C = ( x F y ) ) )
11	10	2rexbidv 2403	. . . . 5 |- ( z = C -> ( E. x e. A E. y e. B z = ( x F y ) <-> E. x e. A E. y e. B C = ( x F y ) ) )
12	11	elabg 2761	. . . 4 |- ( C e. _V -> ( C e. { z | E. x e. A E. y e. B z = ( x F y ) } <-> E. x e. A E. y e. B C = ( x F y ) ) )
13	12	a1i 9	. . 3 |- ( F Fn ( A X. B ) -> ( C e. _V -> ( C e. { z | E. x e. A E. y e. B z = ( x F y ) } <-> E. x e. A E. y e. B C = ( x F y ) ) ) )
14	4, 9, 13	pm5.21ndd 656	. 2 |- ( F Fn ( A X. B ) -> ( C e. { z | E. x e. A E. y e. B z = ( x F y ) } <-> E. x e. A E. y e. B C = ( x F y ) ) )
15	2, 14	bitrd 186	1 |- ( F Fn ( A X. B ) -> ( C e. ran F <-> E. x e. A E. y e. B C = ( x F y ) ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 924   = wceq 1289   e. wcel 1438   {cab 2074   E.wrex 2360   _Vcvv 2619   X. cxp 4436   ran crn 4439   Fn wfn 5010  (class class class)co 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023  df-ov 5655

This theorem is referenced by: (None)