MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ffnov Structured version   Visualization version   Unicode version

Theorem ffnov 6400
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 6049 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F `  w )  e.  C
) )
2 fveq2 5865 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6293 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2503 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eleq1d 2513 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F `
 w )  e.  C  <->  ( x F y )  e.  C
) )
65ralxp 4976 . . 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 700 . 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 253 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 setvar class
Syntax hints:    <-> wb 188    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   <.cop 3974    X. cxp 4832    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fv 5590  df-ov 6293
This theorem is referenced by:  fovcl  6401  cantnfvalf  8170  axaddf  9569  axmulf  9570  mulnzcnopr  10258  frmdplusg  16638  gass  16955  sylow2blem2  17273  matecl  19450  txdis1cn  20650  isxmet2d  21342  prdsmet  21385  imasdsf1olem  21388  imasf1oxmet  21390  imasf1omet  21391  xmetresbl  21452  comet  21528  tgqioo  21818  xrtgioo  21824  opnmblALT  22561  dvdsmulf1o  24123  issubgoi  26038  ghgrpOLD  26096  fovcld  28239  pstmxmet  28700  xrge0pluscn  28746  isbndx  32114  isbnd3  32116  isbnd3b  32117  prdsbnd  32125  isdrngo2  32197  clintopcllaw  39900
  Copyright terms: Public domain W3C validator