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Theorem fnwe2lem1 27139
Description: Lemma for fnwe2 27142. 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 2791 . . 3  |-  ( ph  ->  A. x  e.  A  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
3 fveq2 5731 . . . . . . 7  |-  ( a  =  x  ->  ( F `  a )  =  ( F `  x ) )
43csbeq1d 3259 . . . . . 6  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  x )  /  z ]_ S )
5 fvex 5745 . . . . . . 7  |-  ( F `
 x )  e. 
_V
6 nfcv 2574 . . . . . . 7  |-  F/_ z U
7 fnwe2.su . . . . . . 7  |-  ( z  =  ( F `  x )  ->  S  =  U )
85, 6, 7csbief 3294 . . . . . 6  |-  [_ ( F `  x )  /  z ]_ S  =  U
94, 8syl6eq 2486 . . . . 5  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  U )
103eqeq2d 2449 . . . . . 6  |-  ( a  =  x  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  y )  =  ( F `  x ) ) )
1110rabbidv 2950 . . . . 5  |-  ( a  =  x  ->  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  =  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
129, 11weeq12d 27128 . . . 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 2934 . . 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 205 . 2  |-  ( ph  ->  A. a  e.  A  [_ ( F `  a
)  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
1514r19.21bi 2806 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 359    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711   [_csb 3253   class class class wbr 4215   {copab 4268    We wwe 4543   ` cfv 5457
This theorem is referenced by:  fnwe2lem2  27140  fnwe2lem3  27141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-nul 4341
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-iota 5421  df-fv 5465
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