Theorem ovelrn 5873

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 5870	. . 3 |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )
2	1	eleq2d 2184	. 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 2668	. . . 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 5758	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( x F y ) e. _V )
6		eleq1 2177	. . . . . 6 |- ( C = ( x F y ) -> ( C e. _V <-> ( x F y ) e. _V ) )
7	5, 6	syl5ibrcom 156	. . . . 5 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( C = ( x F y ) -> C e. _V ) )
8	7	3expb 1165	. . . 4 |- ( ( F Fn ( A X. B ) /\ ( x e. A /\ y e. B ) ) -> ( C = ( x F y ) -> C e. _V ) )
9	8	rexlimdvva 2531	. . 3 |- ( F Fn ( A X. B ) -> ( E. x e. A E. y e. B C = ( x F y ) -> C e. _V ) )
10		eqeq1 2121	. . . . . 6 |- ( z = C -> ( z = ( x F y ) <-> C = ( x F y ) ) )
11	10	2rexbidv 2434	. . . . 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 2799	. . . 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 677	. 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 187	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 104   /\ w3a 945   = wceq 1314   e. wcel 1463   {cab 2101   E.wrex 2391   _Vcvv 2657   X. cxp 4497   ran crn 4500   Fn wfn 5076  (class class class)co 5728

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-iota 5046  df-fun 5083  df-fn 5084  df-fv 5089  df-ov 5731

This theorem is referenced by:  blrnps  12400  blrn  12401  tgioo  12532