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Theorem fnwe2val 27146
Description: Lemma for fnwe2 27150. 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 2791 . 2  |-  a  e. 
_V
2 vex 2791 . 2  |-  b  e. 
_V
3 fveq2 5525 . . . 4  |-  ( x  =  a  ->  ( F `  x )  =  ( F `  a ) )
4 fveq2 5525 . . . 4  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
53, 4breqan12d 4038 . . 3  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( F `  x ) R ( F `  y )  <-> 
( F `  a
) R ( F `
 b ) ) )
63, 4eqeqan12d 2298 . . . 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 5539 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
9 nfcv 2419 . . . . . . . 8  |-  F/_ z U
10 fnwe2.su . . . . . . . 8  |-  ( z  =  ( F `  x )  ->  S  =  U )
118, 9, 10csbief 3122 . . . . . . 7  |-  [_ ( F `  x )  /  z ]_ S  =  U
123csbeq1d 3087 . . . . . . 7  |-  ( x  =  a  ->  [_ ( F `  x )  /  z ]_ S  =  [_ ( F `  a )  /  z ]_ S )
1311, 12syl5eqr 2329 . . . . . 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 4037 . . . 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 4282 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 1623   [_csb 3081   class class class wbr 4023   {copab 4076   ` cfv 5255
This theorem is referenced by:  fnwe2lem2  27148  fnwe2lem3  27149
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  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-br 4024  df-opab 4078  df-iota 5219  df-fv 5263
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