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Theorem oemapval 7639
Description: Value of the relation  T. (Contributed by Mario Carneiro, 28-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( 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.3  |-  ( ph  ->  F  e.  S )
oemapval.4  |-  ( 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.3 . 2  |-  ( ph  ->  F  e.  S )
2 oemapval.4 . 2  |-  ( ph  ->  G  e.  S )
3 fveq1 5727 . . . . . 6  |-  ( x  =  F  ->  (
x `  z )  =  ( F `  z ) )
4 fveq1 5727 . . . . . 6  |-  ( y  =  G  ->  (
y `  z )  =  ( G `  z ) )
5 eleq12 2498 . . . . . 6  |-  ( ( ( x `  z
)  =  ( F `
 z )  /\  ( y `  z
)  =  ( G `
 z ) )  ->  ( ( x `
 z )  e.  ( y `  z
)  <->  ( F `  z )  e.  ( G `  z ) ) )
63, 4, 5syl2an 464 . . . . 5  |-  ( ( x  =  F  /\  y  =  G )  ->  ( ( x `  z )  e.  ( y `  z )  <-> 
( F `  z
)  e.  ( G `
 z ) ) )
7 fveq1 5727 . . . . . . . 8  |-  ( x  =  F  ->  (
x `  w )  =  ( F `  w ) )
8 fveq1 5727 . . . . . . . 8  |-  ( y  =  G  ->  (
y `  w )  =  ( G `  w ) )
97, 8eqeqan12d 2451 . . . . . . 7  |-  ( ( x  =  F  /\  y  =  G )  ->  ( ( x `  w )  =  ( y `  w )  <-> 
( F `  w
)  =  ( G `
 w ) ) )
109imbi2d 308 . . . . . 6  |-  ( ( x  =  F  /\  y  =  G )  ->  ( ( z  e.  w  ->  ( x `  w )  =  ( y `  w ) )  <->  ( z  e.  w  ->  ( F `  w )  =  ( G `  w ) ) ) )
1110ralbidv 2725 . . . . 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 692 . . . 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 2726 . . 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 4469 . 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 643 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706   class class class wbr 4212   {copab 4265   Oncon0 4581   dom cdm 4878   ` cfv 5454  (class class class)co 6081   CNF ccnf 7616
This theorem is referenced by:  oemapvali  7640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-iota 5418  df-fv 5462
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