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Theorem fnwe2lem1 27147
Description: Lemma for fnwe2 27150. 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 2626 . . 3  |-  ( ph  ->  A. x  e.  A  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
3 fveq2 5525 . . . . . . 7  |-  ( a  =  x  ->  ( F `  a )  =  ( F `  x ) )
43csbeq1d 3087 . . . . . 6  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  x )  /  z ]_ S )
5 fvex 5539 . . . . . . 7  |-  ( F `
 x )  e. 
_V
6 nfcv 2419 . . . . . . 7  |-  F/_ z U
7 fnwe2.su . . . . . . 7  |-  ( z  =  ( F `  x )  ->  S  =  U )
85, 6, 7csbief 3122 . . . . . 6  |-  [_ ( F `  x )  /  z ]_ S  =  U
94, 8syl6eq 2331 . . . . 5  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  U )
103eqeq2d 2294 . . . . . 6  |-  ( a  =  x  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  y )  =  ( F `  x ) ) )
1110rabbidv 2780 . . . . 5  |-  ( a  =  x  ->  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  =  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
129, 11weeq12d 27136 . . . 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 2764 . . 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 203 . 2  |-  ( ph  ->  A. a  e.  A  [_ ( F `  a
)  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
1514r19.21bi 2641 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 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547   [_csb 3081   class class class wbr 4023   {copab 4076    We wwe 4351   ` 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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  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-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-iota 5219  df-fv 5263
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