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Theorem scottexs 7775
Description: Theorem scheme version of scottex 7773. The collection of all  x of minimum rank such that 
ph ( x ) is true, is a set. (Contributed by NM, 13-Oct-2003.)
Assertion
Ref Expression
scottexs  |-  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }  e.  _V
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem scottexs
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfcv 2548 . . . 4  |-  F/_ z { x  |  ph }
2 nfab1 2550 . . . 4  |-  F/_ x { x  |  ph }
3 nfv 1626 . . . . 5  |-  F/ x
( rank `  z )  C_  ( rank `  y
)
42, 3nfral 2727 . . . 4  |-  F/ x A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )
5 nfv 1626 . . . 4  |-  F/ z A. y  e.  {
x  |  ph } 
( rank `  x )  C_  ( rank `  y
)
6 fveq2 5695 . . . . . 6  |-  ( z  =  x  ->  ( rank `  z )  =  ( rank `  x
) )
76sseq1d 3343 . . . . 5  |-  ( z  =  x  ->  (
( rank `  z )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
87ralbidv 2694 . . . 4  |-  ( z  =  x  ->  ( A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )  <->  A. y  e.  { x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) ) )
91, 2, 4, 5, 8cbvrab 2922 . . 3  |-  { z  e.  { x  | 
ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  =  { x  e. 
{ x  |  ph }  |  A. y  e.  { x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) }
10 df-rab 2683 . . 3  |-  { x  e.  { x  |  ph }  |  A. y  e.  { x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) }  =  { x  |  (
x  e.  { x  |  ph }  /\  A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y ) ) }
11 abid 2400 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
12 df-ral 2679 . . . . . 6  |-  ( A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y )  <->  A. y
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
13 df-sbc 3130 . . . . . . . 8  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
1413imbi1i 316 . . . . . . 7  |-  ( (
[. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) )  <-> 
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1514albii 1572 . . . . . 6  |-  ( A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) )  <->  A. y
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1612, 15bitr4i 244 . . . . 5  |-  ( A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y )  <->  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1711, 16anbi12i 679 . . . 4  |-  ( ( x  e.  { x  |  ph }  /\  A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y ) )  <-> 
( ph  /\  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) ) )
1817abbii 2524 . . 3  |-  { x  |  ( x  e. 
{ x  |  ph }  /\  A. y  e. 
{ x  |  ph }  ( rank `  x
)  C_  ( rank `  y ) ) }  =  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }
199, 10, 183eqtri 2436 . 2  |-  { z  e.  { x  | 
ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  =  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }
20 scottex 7773 . 2  |-  { z  e.  { x  | 
ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  e.  _V
2119, 20eqeltrri 2483 1  |-  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }  e.  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1546    e. wcel 1721   {cab 2398   A.wral 2674   {crab 2678   _Vcvv 2924   [.wsbc 3129    C_ wss 3288   ` cfv 5421   rankcrnk 7653
This theorem is referenced by:  hta  7785
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-reg 7524  ax-inf2 7560
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-recs 6600  df-rdg 6635  df-r1 7654  df-rank 7655
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