Theorem ovelrn 6170

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 6167	. . 3 |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )
2	1	eleq2d 2301	. 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 2814	. . . 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 6050	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( x F y ) e. _V )
6		eleq1 2294	. . . . . 6 |- ( C = ( x F y ) -> ( C e. _V <-> ( x F y ) e. _V ) )
7	5, 6	syl5ibrcom 157	. . . . 5 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( C = ( x F y ) -> C e. _V ) )
8	7	3expb 1230	. . . 4 |- ( ( F Fn ( A X. B ) /\ ( x e. A /\ y e. B ) ) -> ( C = ( x F y ) -> C e. _V ) )
9	8	rexlimdvva 2658	. . 3 |- ( F Fn ( A X. B ) -> ( E. x e. A E. y e. B C = ( x F y ) -> C e. _V ) )
10		eqeq1 2238	. . . . . 6 |- ( z = C -> ( z = ( x F y ) <-> C = ( x F y ) ) )
11	10	2rexbidv 2557	. . . . 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 2952	. . . 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 712	. 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 188	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 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   _Vcvv 2802   X. cxp 4723   ran crn 4726   Fn wfn 5321  (class class class)co 6017

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020

This theorem is referenced by:  blrnps  15134  blrn  15135  tgioo  15277