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Theorem eqfnov 5966
Description: Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)
Assertion
Ref Expression
eqfnov  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( C  X.  D ) )  -> 
( F  =  G  <-> 
( ( A  X.  B )  =  ( C  X.  D )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, F, y    x, G, y
Allowed substitution hints:    C( x, y)    D( x, y)

Proof of Theorem eqfnov
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 eqfnfv2 5639 . 2  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( C  X.  D ) )  -> 
( F  =  G  <-> 
( ( A  X.  B )  =  ( C  X.  D )  /\  A. z  e.  ( A  X.  B
) ( F `  z )  =  ( G `  z ) ) ) )
2 fveq2 5541 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 fveq2 5541 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
42, 3eqeq12d 2310 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  ( G `  z
)  <->  ( F `  <. x ,  y >.
)  =  ( G `
 <. x ,  y
>. ) ) )
5 df-ov 5877 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
6 df-ov 5877 . . . . . 6  |-  ( x G y )  =  ( G `  <. x ,  y >. )
75, 6eqeq12i 2309 . . . . 5  |-  ( ( x F y )  =  ( x G y )  <->  ( F `  <. x ,  y
>. )  =  ( G `  <. x ,  y >. ) )
84, 7syl6bbr 254 . . . 4  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  ( G `  z
)  <->  ( x F y )  =  ( x G y ) ) )
98ralxp 4843 . . 3  |-  ( A. z  e.  ( A  X.  B ) ( F `
 z )  =  ( G `  z
)  <->  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) )
109anbi2i 675 . 2  |-  ( ( ( A  X.  B
)  =  ( C  X.  D )  /\  A. z  e.  ( A  X.  B ) ( F `  z )  =  ( G `  z ) )  <->  ( ( A  X.  B )  =  ( C  X.  D
)  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) )
111, 10syl6bb 252 1  |-  ( ( F  Fn  ( A  X.  B )  /\  G  Fn  ( C  X.  D ) )  -> 
( F  =  G  <-> 
( ( A  X.  B )  =  ( C  X.  D )  /\  A. x  e.  A  A. y  e.  B  ( x F y )  =  ( x G y ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632   A.wral 2556   <.cop 3656    X. cxp 4703    Fn wfn 5266   ` cfv 5271  (class class class)co 5874
This theorem is referenced by:  eqfnov2  5967  ssceq  13719  sspg  21320  ssps  21322  sspmlem  21324
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-13 1698  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-pow 4204  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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877
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