Theorem ffnov 6165

Description: An operation maps to a class to which all values belong. (Contributed by NM, 7-Feb-2004.)

Assertion

Ref	Expression
ffnov	|- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) )

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

Proof of Theorem ffnov

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffnfv 5840	. 2 |- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. w e. ( A X. B ) ( F ` w ) e. C ) )
2		fveq2 5675	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 6061	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2285	. . . . 5 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eleq1d 2303	. . . 4 |- ( w = <. x , y >. -> ( ( F ` w ) e. C <-> ( x F y ) e. C ) )
6	5	ralxp 4903	. . 3 |- ( A. w e. ( A X. B ) ( F ` w ) e. C <-> A. x e. A A. y e. B ( x F y ) e. C )
7	6	anbi2i 457	. 2 |- ( ( F Fn ( A X. B ) /\ A. w e. ( A X. B ) ( F ` w ) e. C ) <-> ( F Fn ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) )
8	1, 7	bitri 184	1 |- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) e. C ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   <.cop 3697   X. cxp 4752   Fn wfn 5352   -->wf 5353   ` cfv 5357  (class class class)co 6058

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061

This theorem is referenced by:  fovcld  6166  axaddf  8199  axmulf  8200  txdis1cn  15269  isxmet2d  15339  xmetresbl  15431  comet  15490  tgqioo  15546  mpodvdsmulf1o  15984