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Theorem scottexs 7746
Description: Theorem scheme version of scottex 7744. 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 2525 . . . 4  |-  F/_ z { x  |  ph }
2 nfab1 2527 . . . 4  |-  F/_ x { x  |  ph }
3 nfv 1626 . . . . 5  |-  F/ x
( rank `  z )  C_  ( rank `  y
)
42, 3nfral 2704 . . . 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 5670 . . . . . 6  |-  ( z  =  x  ->  ( rank `  z )  =  ( rank `  x
) )
76sseq1d 3320 . . . . 5  |-  ( z  =  x  ->  (
( rank `  z )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
87ralbidv 2671 . . . 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 2899 . . 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 2660 . . 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 2377 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
12 df-ral 2656 . . . . . 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 3107 . . . . . . . 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 2501 . . 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 2413 . 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 7744 . 2  |-  { z  e.  { x  | 
ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  e.  _V
2119, 20eqeltrri 2460 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 1717   {cab 2375   A.wral 2651   {crab 2655   _Vcvv 2901   [.wsbc 3106    C_ wss 3265   ` cfv 5396   rankcrnk 7624
This theorem is referenced by:  hta  7756
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-reg 7495  ax-inf2 7531
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-recs 6571  df-rdg 6606  df-r1 7625  df-rank 7626
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