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Theorem fnwe2lem1 27015
Description: Lemma for fnwe2 27018. Substitution in well-ordering hypothesis. (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 ) ) }
fnwe2.s  |-  ( (
ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
Assertion
Ref Expression
fnwe2lem1  |-  ( (
ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
Distinct variable groups:    y, U, z, a    x, S, y, a    x, R, y, a    ph, x, y, z   
x, A, y, z, a    x, F, y, z, a    T, a
Allowed substitution hints:    ph( a)    R( z)    S( z)    T( x, y, z)    U( x)

Proof of Theorem fnwe2lem1
StepHypRef Expression
1 fnwe2.s . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
21ralrimiva 2749 . . 3  |-  ( ph  ->  A. x  e.  A  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
3 fveq2 5687 . . . . . . 7  |-  ( a  =  x  ->  ( F `  a )  =  ( F `  x ) )
43csbeq1d 3217 . . . . . 6  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  x )  /  z ]_ S )
5 fvex 5701 . . . . . . 7  |-  ( F `
 x )  e. 
_V
6 nfcv 2540 . . . . . . 7  |-  F/_ z U
7 fnwe2.su . . . . . . 7  |-  ( z  =  ( F `  x )  ->  S  =  U )
85, 6, 7csbief 3252 . . . . . 6  |-  [_ ( F `  x )  /  z ]_ S  =  U
94, 8syl6eq 2452 . . . . 5  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  U )
103eqeq2d 2415 . . . . . 6  |-  ( a  =  x  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  y )  =  ( F `  x ) ) )
1110rabbidv 2908 . . . . 5  |-  ( a  =  x  ->  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  =  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
129, 11weeq12d 27004 . . . 4  |-  ( a  =  x  ->  ( [_ ( F `  a
)  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  <->  U  We  { y  e.  A  | 
( F `  y
)  =  ( F `
 x ) } ) )
1312cbvralv 2892 . . 3  |-  ( A. a  e.  A  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  <->  A. x  e.  A  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
142, 13sylibr 204 . 2  |-  ( ph  ->  A. a  e.  A  [_ ( F `  a
)  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
1514r19.21bi 2764 1  |-  ( (
ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   {crab 2670   [_csb 3211   class class class wbr 4172   {copab 4225    We wwe 4500   ` cfv 5413
This theorem is referenced by:  fnwe2lem2  27016  fnwe2lem3  27017
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-nul 4298
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-iota 5377  df-fv 5421
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