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Theorem fnwe2val 35338
Description: Lemma for fnwe2 35342. 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 3061 . 2  |-  a  e. 
_V
2 vex 3061 . 2  |-  b  e. 
_V
3 fveq2 5805 . . . 4  |-  ( x  =  a  ->  ( F `  x )  =  ( F `  a ) )
4 fveq2 5805 . . . 4  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
53, 4breqan12d 4409 . . 3  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( F `  x ) R ( F `  y )  <-> 
( F `  a
) R ( F `
 b ) ) )
63, 4eqeqan12d 2425 . . . 4  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
( F `  a
)  =  ( F `
 b ) ) )
7 simpl 455 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  x  =  a )
8 fvex 5815 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
9 fnwe2.su . . . . . . . 8  |-  ( z  =  ( F `  x )  ->  S  =  U )
108, 9csbie 3398 . . . . . . 7  |-  [_ ( F `  x )  /  z ]_ S  =  U
113csbeq1d 3379 . . . . . . 7  |-  ( x  =  a  ->  [_ ( F `  x )  /  z ]_ S  =  [_ ( F `  a )  /  z ]_ S )
1210, 11syl5eqr 2457 . . . . . 6  |-  ( x  =  a  ->  U  =  [_ ( F `  a )  /  z ]_ S )
1312adantr 463 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  U  =  [_ ( F `  a )  /  z ]_ S
)
14 simpr 459 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  y  =  b )
157, 13, 14breq123d 4408 . . . 4  |-  ( ( x  =  a  /\  y  =  b )  ->  ( x U y  <-> 
a [_ ( F `  a )  /  z ]_ S b ) )
166, 15anbi12d 709 . . 3  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( ( F `
 x )  =  ( F `  y
)  /\  x U
y )  <->  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a
)  /  z ]_ S b ) ) )
175, 16orbi12d 708 . 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 ) ) ) )
18 fnwe2.t . 2  |-  T  =  { <. x ,  y
>.  |  ( ( F `  x ) R ( F `  y )  \/  (
( F `  x
)  =  ( F `
 y )  /\  x U y ) ) }
191, 2, 17, 18braba 4706 1  |-  ( a T b  <->  ( ( F `  a ) R ( F `  b )  \/  (
( F `  a
)  =  ( F `
 b )  /\  a [_ ( F `  a )  /  z ]_ S b ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405   [_csb 3372   class class class wbr 4394   {copab 4451   ` cfv 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-iota 5489  df-fv 5533
This theorem is referenced by:  fnwe2lem2  35340  fnwe2lem3  35341
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