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Theorem fnwe2val 29573
Description: Lemma for fnwe2 29577. 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 3081 . 2  |-  a  e. 
_V
2 vex 3081 . 2  |-  b  e. 
_V
3 fveq2 5802 . . . 4  |-  ( x  =  a  ->  ( F `  x )  =  ( F `  a ) )
4 fveq2 5802 . . . 4  |-  ( y  =  b  ->  ( F `  y )  =  ( F `  b ) )
53, 4breqan12d 4418 . . 3  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( F `  x ) R ( F `  y )  <-> 
( F `  a
) R ( F `
 b ) ) )
63, 4eqeqan12d 2477 . . . 4  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( F `  x )  =  ( F `  y )  <-> 
( F `  a
)  =  ( F `
 b ) ) )
7 simpl 457 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  x  =  a )
8 fvex 5812 . . . . . . . 8  |-  ( F `
 x )  e. 
_V
9 nfcv 2616 . . . . . . . 8  |-  F/_ z U
10 fnwe2.su . . . . . . . 8  |-  ( z  =  ( F `  x )  ->  S  =  U )
118, 9, 10csbief 3423 . . . . . . 7  |-  [_ ( F `  x )  /  z ]_ S  =  U
123csbeq1d 3405 . . . . . . 7  |-  ( x  =  a  ->  [_ ( F `  x )  /  z ]_ S  =  [_ ( F `  a )  /  z ]_ S )
1311, 12syl5eqr 2509 . . . . . 6  |-  ( x  =  a  ->  U  =  [_ ( F `  a )  /  z ]_ S )
1413adantr 465 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  U  =  [_ ( F `  a )  /  z ]_ S
)
15 simpr 461 . . . . 5  |-  ( ( x  =  a  /\  y  =  b )  ->  y  =  b )
167, 14, 15breq123d 4417 . . . 4  |-  ( ( x  =  a  /\  y  =  b )  ->  ( x U y  <-> 
a [_ ( F `  a )  /  z ]_ S b ) )
176, 16anbi12d 710 . . 3  |-  ( ( x  =  a  /\  y  =  b )  ->  ( ( ( F `
 x )  =  ( F `  y
)  /\  x U
y )  <->  ( ( F `  a )  =  ( F `  b )  /\  a [_ ( F `  a
)  /  z ]_ S b ) ) )
185, 17orbi12d 709 . 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 4717 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 368    /\ wa 369    = wceq 1370   [_csb 3398   class class class wbr 4403   {copab 4460   ` cfv 5529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-iota 5492  df-fv 5537
This theorem is referenced by:  fnwe2lem2  29575  fnwe2lem3  29576
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