Theorem ffnov 6108

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 5793	. 2 |- ( F : ( A X. B ) --> C <-> ( F Fn ( A X. B ) /\ A. w e. ( A X. B ) ( F ` w ) e. C ) )
2		fveq2 5627	. . . . . 6 |- ( w = <. x , y >. -> ( F ` w ) = ( F ` <. x , y >. ) )
3		df-ov 6004	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2280	. . . . 5 |- ( w = <. x , y >. -> ( F ` w ) = ( x F y ) )
5	4	eleq1d 2298	. . . 4 |- ( w = <. x , y >. -> ( ( F ` w ) e. C <-> ( x F y ) e. C ) )
6	5	ralxp 4865	. . 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 1395   e. wcel 2200   A.wral 2508   <.cop 3669   X. cxp 4717   Fn wfn 5313   -->wf 5314   ` cfv 5318  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004

This theorem is referenced by:  fovcld  6109  axaddf  8055  axmulf  8056  txdis1cn  14952  isxmet2d  15022  xmetresbl  15114  comet  15173  tgqioo  15229  mpodvdsmulf1o  15664