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Theorem scottexf 32381
Description: A version of scottex 8370 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 2585 . . . . . 6  |-  F/_ z A
3 nfv 1756 . . . . . 6  |-  F/ z ( rank `  x
)  C_  ( rank `  y )
4 nfv 1756 . . . . . 6  |-  F/ y ( rank `  x
)  C_  ( rank `  z )
5 fveq2 5887 . . . . . . 7  |-  ( y  =  z  ->  ( rank `  y )  =  ( rank `  z
) )
65sseq2d 3498 . . . . . 6  |-  ( y  =  z  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
71, 2, 3, 4, 6cbvralf 3053 . . . . 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 3073 . . 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 2585 . . . 4  |-  F/_ w A
11 scottexf.2 . . . 4  |-  F/_ x A
12 nfv 1756 . . . . 5  |-  F/ x
( rank `  w )  C_  ( rank `  z
)
1311, 12nfral 2813 . . . 4  |-  F/ x A. z  e.  A  ( rank `  w )  C_  ( rank `  z
)
14 nfv 1756 . . . 4  |-  F/ w A. z  e.  A  ( rank `  x )  C_  ( rank `  z
)
15 fveq2 5887 . . . . . 6  |-  ( w  =  x  ->  ( rank `  w )  =  ( rank `  x
) )
1615sseq1d 3497 . . . . 5  |-  ( w  =  x  ->  (
( rank `  w )  C_  ( rank `  z
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
1716ralbidv 2866 . . . 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 3083 . . 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 2455 . 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 8370 . 2  |-  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z ) }  e.  _V
2119, 20eqeltri 2508 1  |-  { x  e.  A  |  A. y  e.  A  ( rank `  x )  C_  ( rank `  y ) }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1438    e. wcel 1873   F/_wnfc 2571   A.wral 2776   {crab 2780   _Vcvv 3085    C_ wss 3442   ` cfv 5607   rankcrnk 8248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-8 1875  ax-9 1877  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402  ax-rep 4542  ax-sep 4552  ax-nul 4561  ax-pow 4608  ax-pr 4666  ax-un 6603  ax-reg 8122  ax-inf2 8161
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3087  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3918  df-pw 3989  df-sn 4005  df-pr 4007  df-tp 4009  df-op 4011  df-uni 4226  df-int 4262  df-iun 4307  df-br 4430  df-opab 4489  df-mpt 4490  df-tr 4525  df-eprel 4770  df-id 4774  df-po 4780  df-so 4781  df-fr 4818  df-we 4820  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-pred 5405  df-ord 5451  df-on 5452  df-lim 5453  df-suc 5454  df-iota 5571  df-fun 5609  df-fn 5610  df-f 5611  df-f1 5612  df-fo 5613  df-f1o 5614  df-fv 5615  df-om 6713  df-wrecs 7045  df-recs 7107  df-rdg 7145  df-r1 8249  df-rank 8250
This theorem is referenced by: (None)
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