ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ffnov Unicode version

Theorem ffnov 5925
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.
StepHypRef Expression
1 ffnfv 5625 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F `  w )  e.  C
) )
2 fveq2 5468 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5827 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3eqtr4di 2208 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eleq1d 2226 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F `
 w )  e.  C  <->  ( x F y )  e.  C
) )
65ralxp 4729 . . 3  |-  ( A. w  e.  ( A  X.  B ) ( F `
 w )  e.  C  <->  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C )
76anbi2i 453 . 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 ) )
81, 7bitri 183 1  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. x  e.  A  A. y  e.  B  ( x F y )  e.  C ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   A.wral 2435   <.cop 3563    X. cxp 4584    Fn wfn 5165   -->wf 5166   ` cfv 5170  (class class class)co 5824
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-ov 5827
This theorem is referenced by:  fovcl  5926  axaddf  7788  axmulf  7789  txdis1cn  12689  isxmet2d  12759  xmetresbl  12851  comet  12910  tgqioo  12958
  Copyright terms: Public domain W3C validator