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Theorem fnwe2lem1 30924
Description: Lemma for fnwe2 30927. 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 2881 . . 3  |-  ( ph  ->  A. x  e.  A  U  We  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
3 fveq2 5872 . . . . . . 7  |-  ( a  =  x  ->  ( F `  a )  =  ( F `  x ) )
43csbeq1d 3447 . . . . . 6  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  [_ ( F `  x )  /  z ]_ S )
5 fvex 5882 . . . . . . 7  |-  ( F `
 x )  e. 
_V
6 nfcv 2629 . . . . . . 7  |-  F/_ z U
7 fnwe2.su . . . . . . 7  |-  ( z  =  ( F `  x )  ->  S  =  U )
85, 6, 7csbief 3465 . . . . . 6  |-  [_ ( F `  x )  /  z ]_ S  =  U
94, 8syl6eq 2524 . . . . 5  |-  ( a  =  x  ->  [_ ( F `  a )  /  z ]_ S  =  U )
103eqeq2d 2481 . . . . . 6  |-  ( a  =  x  ->  (
( F `  y
)  =  ( F `
 a )  <->  ( F `  y )  =  ( F `  x ) ) )
1110rabbidv 3110 . . . . 5  |-  ( a  =  x  ->  { y  e.  A  |  ( F `  y )  =  ( F `  a ) }  =  { y  e.  A  |  ( F `  y )  =  ( F `  x ) } )
129, 11weeq12d 30913 . . . 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 3093 . . 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 212 . 2  |-  ( ph  ->  A. a  e.  A  [_ ( F `  a
)  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
1514r19.21bi 2836 1  |-  ( (
ph  /\  a  e.  A )  ->  [_ ( F `  a )  /  z ]_ S  We  { y  e.  A  |  ( F `  y )  =  ( F `  a ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821   [_csb 3440   class class class wbr 4453   {copab 4510    We wwe 4843   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-nul 4582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-iota 5557  df-fv 5602
This theorem is referenced by:  fnwe2lem2  30925  fnwe2lem3  30926
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