Theorem ovelrn 6069

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 6066	. . 3 |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )
2	1	eleq2d 2263	. 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 2771	. . . 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 5952	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( x F y ) e. _V )
6		eleq1 2256	. . . . . 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 1206	. . . 4 |- ( ( F Fn ( A X. B ) /\ ( x e. A /\ y e. B ) ) -> ( C = ( x F y ) -> C e. _V ) )
9	8	rexlimdvva 2619	. . 3 |- ( F Fn ( A X. B ) -> ( E. x e. A E. y e. B C = ( x F y ) -> C e. _V ) )
10		eqeq1 2200	. . . . . 6 |- ( z = C -> ( z = ( x F y ) <-> C = ( x F y ) ) )
11	10	2rexbidv 2519	. . . . 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 2907	. . . 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 706	. 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 980   = wceq 1364   e. wcel 2164   {cab 2179   E.wrex 2473   _Vcvv 2760   X. cxp 4658   ran crn 4661   Fn wfn 5250  (class class class)co 5919

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-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922

This theorem is referenced by:  blrnps  14590  blrn  14591  tgioo  14733