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Theorem ffnov 6176
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 5896 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F `  w )  e.  C
) )
2 fveq2 5730 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 6086 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2488 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eleq1d 2504 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F `
 w )  e.  C  <->  ( x F y )  e.  C
) )
65ralxp 5018 . . 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 677 . 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 242 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:    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   <.cop 3819    X. cxp 4878    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083
This theorem is referenced by:  fovcl  6177  cantnfvalf  7622  axaddf  9022  axmulf  9023  mulnzcnopr  9670  mndfo  14722  frmdplusg  14801  gass  15080  sylow2blem2  15257  txdis1cn  17669  isxmet2d  18359  prdsmet  18402  imasdsf1olem  18405  imasf1oxmet  18407  imasf1omet  18408  xmetresbl  18469  comet  18545  tgqioo  18833  xrtgioo  18839  opnmblALT  19497  dvdsmulf1o  20981  issubgoi  21900  ghgrp  21958  fovcld  24052  ofrn  24054  pstmxmet  24294  xrge0pluscn  24328  isbndx  26493  isbnd3  26495  isbnd3b  26496  prdsbnd  26504  isdrngo2  26576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086
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