Theorem ovelrn 6097

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 6094	. . 3 |- ( F Fn ( A X. B ) -> ran F = { z | E. x e. A E. y e. B z = ( x F y ) } )
2	1	eleq2d 2275	. 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 2783	. . . 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 5979	. . . . . 6 |- ( ( F Fn ( A X. B ) /\ x e. A /\ y e. B ) -> ( x F y ) e. _V )
6		eleq1 2268	. . . . . 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 1207	. . . 4 |- ( ( F Fn ( A X. B ) /\ ( x e. A /\ y e. B ) ) -> ( C = ( x F y ) -> C e. _V ) )
9	8	rexlimdvva 2631	. . 3 |- ( F Fn ( A X. B ) -> ( E. x e. A E. y e. B C = ( x F y ) -> C e. _V ) )
10		eqeq1 2212	. . . . . 6 |- ( z = C -> ( z = ( x F y ) <-> C = ( x F y ) ) )
11	10	2rexbidv 2531	. . . . 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 2919	. . . 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 707	. 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 981   = wceq 1373   e. wcel 2176   {cab 2191   E.wrex 2485   _Vcvv 2772   X. cxp 4674   ran crn 4677   Fn wfn 5267  (class class class)co 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949

This theorem is referenced by:  blrnps  14916  blrn  14917  tgioo  15059