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Theorem scottexs 8082
Description: Theorem scheme version of scottex 8080. 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 2569 . . . 4  |-  F/_ z { x  |  ph }
2 nfab1 2571 . . . 4  |-  F/_ x { x  |  ph }
3 nfv 1672 . . . . 5  |-  F/ x
( rank `  z )  C_  ( rank `  y
)
42, 3nfral 2759 . . . 4  |-  F/ x A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y )
5 nfv 1672 . . . 4  |-  F/ z A. y  e.  {
x  |  ph } 
( rank `  x )  C_  ( rank `  y
)
6 fveq2 5679 . . . . . 6  |-  ( z  =  x  ->  ( rank `  z )  =  ( rank `  x
) )
76sseq1d 3371 . . . . 5  |-  ( z  =  x  ->  (
( rank `  z )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  y ) ) )
87ralbidv 2725 . . . 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 2960 . . 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 2714 . . 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 2421 . . . . 5  |-  ( x  e.  { x  | 
ph }  <->  ph )
12 df-ral 2710 . . . . . 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 3176 . . . . . . . 8  |-  ( [. y  /  x ]. ph  <->  y  e.  { x  |  ph }
)
1413imbi1i 325 . . . . . . 7  |-  ( (
[. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) )  <-> 
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1514albii 1613 . . . . . 6  |-  ( A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) )  <->  A. y
( y  e.  {
x  |  ph }  ->  ( rank `  x
)  C_  ( rank `  y ) ) )
1612, 15bitr4i 252 . . . . 5  |-  ( A. y  e.  { x  |  ph }  ( rank `  x )  C_  ( rank `  y )  <->  A. y
( [. y  /  x ]. ph  ->  ( rank `  x )  C_  ( rank `  y ) ) )
1711, 16anbi12i 690 . . . 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 2545 . . 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 2457 . 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 8080 . 2  |-  { z  e.  { x  | 
ph }  |  A. y  e.  { x  |  ph }  ( rank `  z )  C_  ( rank `  y ) }  e.  _V
2119, 20eqeltrri 2504 1  |-  { x  |  ( ph  /\  A. y ( [. y  /  x ]. ph  ->  (
rank `  x )  C_  ( rank `  y
) ) ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1360    e. wcel 1755   {cab 2419   A.wral 2705   {crab 2709   _Vcvv 2962   [.wsbc 3175    C_ wss 3316   ` cfv 5406   rankcrnk 7958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-reg 7795  ax-inf2 7835
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-om 6466  df-recs 6818  df-rdg 6852  df-r1 7959  df-rank 7960
This theorem is referenced by:  hta  8092
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