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Theorem fnwe2val 27249
Description: Lemma for fnwe2 27253. Substitute variables. (Contributed by Stefan O'Rear, 19-Jan-2015.)
Hypotheses
Ref Expression
fnwe2.su  |-  ( z  =  ( F `  x )  ->  S  =  U )
fnwe2.t  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
Assertion
Ref Expression
fnwe2val  |-  ( a T b  <->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
Distinct variable groups:    y, U, z, a, b    x, S, y, a, b    x, R, y, a, b    x, z, F, y, a, b    T, a, b
Allowed substitution hints:    R( z)    S( z)    T( x, y, z)    U( x)

Proof of Theorem fnwe2val
StepHypRef Expression
1 vex 2804 . 2  |-  a  e. 
_V
2 vex 2804 . 2  |-  b  e. 
_V
3 fveq2 5541 . . . 4  |-  ( x  =  a  ->  ( F `  x )  =  ( F `  a ) )
4 fveq2 5541 . . . 4  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
53, 4breqan12d 4054 . . 3  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( F `  x ) R ( F `  y )  <-> 
( F `  a
) R ( F `
 b ) ) )
63, 4eqeqan12d 2311 . . . 4  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
( F `  a
)  =  ( F `
 b ) ) )
7 simpl 443 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  x  =  a )
8 fvex 5555 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
9 nfcv 2432 . . . . . . . 8  |-  F/_ z U
10 fnwe2.su . . . . . . . 8  |-  ( z  =  ( F `  x )  ->  S  =  U )
118, 9, 10csbief 3135 . . . . . . 7  |-  [_ ( F `  x )  /  z ]_ S  =  U
123csbeq1d 3100 . . . . . . 7  |-  ( x  =  a  ->  [_ ( F `  x )  /  z ]_ S  =  [_ ( F `  a )  /  z ]_ S )
1311, 12syl5eqr 2342 . . . . . 6  |-  ( x  =  a  ->  U  =  [_ ( F `  a )  /  z ]_ S )
1413adantr 451 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  U  =  [_ ( F `  a )  /  z ]_ S
)
15 simpr 447 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  y  =  b )
167, 14, 15breq123d 4053 . . . 4  |-  ( ( x  =  a  /\  y  =  b )  ->  ( x U y  <-> 
a [_ ( F `  a )  /  z ]_ S b ) )
176, 16anbi12d 691 . . 3  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( ( F `
 x )  =  ( F `  y
)  /\  x U
y )  <->  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a
)  /  z ]_ S b ) ) )
185, 17orbi12d 690 . 2  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( ( F `
 x ) R ( F `  y
)  \/  ( ( F `  x )  =  ( F `  y )  /\  x U y ) )  <-> 
( ( F `  a ) R ( F `  b )  \/  ( ( F `
 a )  =  ( F `  b
)  /\  a [_ ( F `  a )  /  z ]_ S
b ) ) ) )
19 fnwe2.t . 2  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
201, 2, 18, 19braba 4298 1  |-  ( a T b  <->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632   [_csb 3094   class class class wbr 4039   {copab 4092   ` cfv 5271
This theorem is referenced by:  fnwe2lem2  27251  fnwe2lem3  27252
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-br 4040  df-opab 4094  df-iota 5235  df-fv 5279
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