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Theorem ffnov 5964
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 5701 . 2  |-  ( F : ( A  X.  B ) --> C  <->  ( F  Fn  ( A  X.  B
)  /\  A. w  e.  ( A  X.  B
) ( F `  w )  e.  C
) )
2 fveq2 5541 . . . . . 6  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( F `  <. x ,  y >. )
)
3 df-ov 5877 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
42, 3syl6eqr 2346 . . . . 5  |-  ( w  =  <. x ,  y
>.  ->  ( F `  w )  =  ( x F y ) )
54eleq1d 2362 . . . 4  |-  ( w  =  <. x ,  y
>.  ->  ( ( F `
 w )  e.  C  <->  ( x F y )  e.  C
) )
65ralxp 4843 . . 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 675 . 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 240 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 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   <.cop 3656    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874
This theorem is referenced by:  fovcl  5965  cantnfvalf  7382  axaddf  8783  axmulf  8784  mulnzcnopr  9430  mndfo  14413  frmdplusg  14492  gass  14771  sylow2blem2  14948  txdis1cn  17345  isxmet2d  17908  prdsmet  17950  imasdsf1olem  17953  imasf1oxmet  17955  imasf1omet  17956  xmetresbl  17999  comet  18075  tgqioo  18322  xrtgioo  18328  opnmblALT  18974  dvdsmulf1o  20450  issubgoi  20993  ghgrp  21051  fovcld  23218  ofrn  23221  xrge0pluscn  23337  isbndx  26609  isbnd3  26611  isbnd3b  26612  prdsbnd  26620  isdrngo2  26692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-ov 5877
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