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Theorem oemapval 8093
Description: Value of the relation  T. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
oemapval.f  |-  ( ph  ->  F  e.  S )
oemapval.g  |-  ( ph  ->  G  e.  S )
Assertion
Ref Expression
oemapval  |-  ( ph  ->  ( F T G  <->  E. z  e.  B  ( ( F `  z )  e.  ( G `  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) ) )
Distinct variable groups:    x, w, y, z, B    w, A, x, y, z    w, F, x, y, z    x, S, y, z    w, G, x, y, z    ph, x, y, z
Allowed substitution hints:    ph( w)    S( w)    T( x, y, z, w)

Proof of Theorem oemapval
StepHypRef Expression
1 oemapval.f . 2  |-  ( ph  ->  F  e.  S )
2 oemapval.g . 2  |-  ( ph  ->  G  e.  S )
3 fveq1 5858 . . . . . 6  |-  ( x  =  F  ->  (
x `  z )  =  ( F `  z ) )
4 fveq1 5858 . . . . . 6  |-  ( y  =  G  ->  (
y `  z )  =  ( G `  z ) )
5 eleq12 2538 . . . . . 6  |-  ( ( ( x `  z
)  =  ( F `
 z )  /\  ( y `  z
)  =  ( G `
 z ) )  ->  ( ( x `
 z )  e.  ( y `  z
)  <->  ( F `  z )  e.  ( G `  z ) ) )
63, 4, 5syl2an 477 . . . . 5  |-  ( ( x  =  F  /\  y  =  G )  ->  ( ( x `  z )  e.  ( y `  z )  <-> 
( F `  z
)  e.  ( G `
 z ) ) )
7 fveq1 5858 . . . . . . . 8  |-  ( x  =  F  ->  (
x `  w )  =  ( F `  w ) )
8 fveq1 5858 . . . . . . . 8  |-  ( y  =  G  ->  (
y `  w )  =  ( G `  w ) )
97, 8eqeqan12d 2485 . . . . . . 7  |-  ( ( x  =  F  /\  y  =  G )  ->  ( ( x `  w )  =  ( y `  w )  <-> 
( F `  w
)  =  ( G `
 w ) ) )
109imbi2d 316 . . . . . 6  |-  ( ( x  =  F  /\  y  =  G )  ->  ( ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) )  <->  ( z  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) )
1110ralbidv 2898 . . . . 5  |-  ( ( x  =  F  /\  y  =  G )  ->  ( A. w  e.  B  ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) )  <->  A. w  e.  B  ( z  e.  w  ->  ( F `  w
)  =  ( G `
 w ) ) ) )
126, 11anbi12d 710 . . . 4  |-  ( ( x  =  F  /\  y  =  G )  ->  ( ( ( x `
 z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) )  <->  ( ( F `  z )  e.  ( G `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( F `
 w )  =  ( G `  w
) ) ) ) )
1312rexbidv 2968 . . 3  |-  ( ( x  =  F  /\  y  =  G )  ->  ( E. z  e.  B  ( ( x `
 z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) )  <->  E. z  e.  B  ( ( F `  z )  e.  ( G `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( F `
 w )  =  ( G `  w
) ) ) ) )
14 oemapval.t . . 3  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
1513, 14brabga 4756 . 2  |-  ( ( F  e.  S  /\  G  e.  S )  ->  ( F T G  <->  E. z  e.  B  ( ( F `  z )  e.  ( G `  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) ) )
161, 2, 15syl2anc 661 1  |-  ( ph  ->  ( F T G  <->  E. z  e.  B  ( ( F `  z )  e.  ( G `  z )  /\  A. w  e.  B  ( z  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2809   E.wrex 2810   class class class wbr 4442   {copab 4499   Oncon0 4873   dom cdm 4994   ` cfv 5581  (class class class)co 6277   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-iota 5544  df-fv 5589
This theorem is referenced by:  oemapvali  8094
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