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Theorem scottexf 29115
Description: A version of scottex 8190 with non-free variables instead of distinct variables. (Contributed by Giovanni Mascellani, 19-Aug-2018.)
Hypotheses
Ref Expression
scottexf.1  |-  F/_ y A
scottexf.2  |-  F/_ x A
Assertion
Ref Expression
scottexf  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)

Proof of Theorem scottexf
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scottexf.1 . . . . . 6  |-  F/_ y A
2 nfcv 2611 . . . . . 6  |-  F/_ z A
3 nfv 1674 . . . . . 6  |-  F/ z ( rank `  x
)  C_  ( rank `  y )
4 nfv 1674 . . . . . 6  |-  F/ y ( rank `  x
)  C_  ( rank `  z )
5 fveq2 5786 . . . . . . 7  |-  ( y  =  z  ->  ( rank `  y )  =  ( rank `  z
) )
65sseq2d 3479 . . . . . 6  |-  ( y  =  z  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
71, 2, 3, 4, 6cbvralf 3034 . . . . 5  |-  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y )  <->  A. z  e.  A  (
rank `  x )  C_  ( rank `  z
) )
87a1i 11 . . . 4  |-  ( x  e.  A  ->  ( A. y  e.  A  ( rank `  x )  C_  ( rank `  y
)  <->  A. z  e.  A  ( rank `  x )  C_  ( rank `  z
) ) )
98rabbiia 3054 . . 3  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  { x  e.  A  |  A. z  e.  A  ( rank `  x )  C_  ( rank `  z ) }
10 nfcv 2611 . . . 4  |-  F/_ w A
11 scottexf.2 . . . 4  |-  F/_ x A
12 nfv 1674 . . . . 5  |-  F/ x
( rank `  w )  C_  ( rank `  z
)
1311, 12nfral 2875 . . . 4  |-  F/ x A. z  e.  A  ( rank `  w )  C_  ( rank `  z
)
14 nfv 1674 . . . 4  |-  F/ w A. z  e.  A  ( rank `  x )  C_  ( rank `  z
)
15 fveq2 5786 . . . . . 6  |-  ( w  =  x  ->  ( rank `  w )  =  ( rank `  x
) )
1615sseq1d 3478 . . . . 5  |-  ( w  =  x  ->  (
( rank `  w )  C_  ( rank `  z
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
1716ralbidv 2839 . . . 4  |-  ( w  =  x  ->  ( A. z  e.  A  ( rank `  w )  C_  ( rank `  z
)  <->  A. z  e.  A  ( rank `  x )  C_  ( rank `  z
) ) )
1810, 11, 13, 14, 17cbvrab 3063 . . 3  |-  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z ) }  =  { x  e.  A  |  A. z  e.  A  ( rank `  x )  C_  ( rank `  z ) }
199, 18eqtr4i 2482 . 2  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  =  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z ) }
20 scottex 8190 . 2  |-  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z ) }  e.  _V
2119, 20eqeltri 2533 1  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   F/_wnfc 2597   A.wral 2793   {crab 2797   _Vcvv 3065    C_ wss 3423   ` cfv 5513   rankcrnk 8068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-reg 7905  ax-inf2 7945
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-om 6574  df-recs 6929  df-rdg 6963  df-r1 8069  df-rank 8070
This theorem is referenced by: (None)
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