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

Theorem eqfnov 5950
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 5623 . 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 5525 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( F `  z )  =  ( F `  <. x ,  y >. )
)
3 fveq2 5525 . . . . . 6  |-  ( z  =  <. x ,  y
>.  ->  ( G `  z )  =  ( G `  <. x ,  y >. )
)
42, 3eqeq12d 2297 . . . . 5  |-  ( z  =  <. x ,  y
>.  ->  ( ( F `
 z )  =  ( G `  z
)  <->  ( F `  <. x ,  y >.
)  =  ( G `
 <. x ,  y
>. ) ) )
5 df-ov 5861 . . . . . 6  |-  ( x F y )  =  ( F `  <. x ,  y >. )
6 df-ov 5861 . . . . . 6  |-  ( x G y )  =  ( G `  <. x ,  y >. )
75, 6eqeq12i 2296 . . . . 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 4827 . . 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 1623   A.wral 2543   <.cop 3643    X. cxp 4687    Fn wfn 5250   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  eqfnov2  5951  ssceq  13703  sspg  21304  ssps  21306  sspmlem  21308
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861
  Copyright terms: Public domain W3C validator