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Theorem scottexf 30178
Description: A version of scottex 8299 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 2629 . . . . . 6  |-  F/_ z A
3 nfv 1683 . . . . . 6  |-  F/ z ( rank `  x
)  C_  ( rank `  y )
4 nfv 1683 . . . . . 6  |-  F/ y ( rank `  x
)  C_  ( rank `  z )
5 fveq2 5864 . . . . . . 7  |-  ( y  =  z  ->  ( rank `  y )  =  ( rank `  z
) )
65sseq2d 3532 . . . . . 6  |-  ( y  =  z  ->  (
( rank `  x )  C_  ( rank `  y
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
71, 2, 3, 4, 6cbvralf 3082 . . . . 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 3102 . . 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 2629 . . . 4  |-  F/_ w A
11 scottexf.2 . . . 4  |-  F/_ x A
12 nfv 1683 . . . . 5  |-  F/ x
( rank `  w )  C_  ( rank `  z
)
1311, 12nfral 2850 . . . 4  |-  F/ x A. z  e.  A  ( rank `  w )  C_  ( rank `  z
)
14 nfv 1683 . . . 4  |-  F/ w A. z  e.  A  ( rank `  x )  C_  ( rank `  z
)
15 fveq2 5864 . . . . . 6  |-  ( w  =  x  ->  ( rank `  w )  =  ( rank `  x
) )
1615sseq1d 3531 . . . . 5  |-  ( w  =  x  ->  (
( rank `  w )  C_  ( rank `  z
)  <->  ( rank `  x
)  C_  ( rank `  z ) ) )
1716ralbidv 2903 . . . 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 3111 . . 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 2499 . 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 8299 . 2  |-  { w  e.  A  |  A. z  e.  A  ( rank `  w )  C_  ( rank `  z ) }  e.  _V
2119, 20eqeltri 2551 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 1379    e. wcel 1767   F/_wnfc 2615   A.wral 2814   {crab 2818   _Vcvv 3113    C_ wss 3476   ` cfv 5586   rankcrnk 8177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-reg 8014  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-recs 7039  df-rdg 7073  df-r1 8178  df-rank 8179
This theorem is referenced by: (None)
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