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Theorem cotrintab 36267
Description: The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)
Hypothesis
Ref Expression
cotrintab.min  |-  ( ph  ->  ( x  o.  x
)  C_  x )
Assertion
Ref Expression
cotrintab  |-  ( |^| { x  |  ph }  o.  |^| { x  | 
ph } )  C_  |^|
{ x  |  ph }

Proof of Theorem cotrintab
Dummy variables  v  u  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cotr 5234 . 2  |-  ( (
|^| { x  |  ph }  o.  |^| { x  |  ph } )  C_  |^|
{ x  |  ph } 
<-> 
A. u A. w A. v ( ( u
|^| { x  |  ph } w  /\  w |^| { x  |  ph } v )  ->  u |^| { x  | 
ph } v ) )
2 pm3.43 878 . . . . . 6  |-  ( ( ( ph  ->  u x w )  /\  ( ph  ->  w x
v ) )  -> 
( ph  ->  ( u x w  /\  w x v ) ) )
3 cotrintab.min . . . . . . 7  |-  ( ph  ->  ( x  o.  x
)  C_  x )
4 cotr 5234 . . . . . . . 8  |-  ( ( x  o.  x ) 
C_  x  <->  A. u A. w A. v ( ( u x w  /\  w x v )  ->  u x
v ) )
54biimpi 199 . . . . . . 7  |-  ( ( x  o.  x ) 
C_  x  ->  A. u A. w A. v ( ( u x w  /\  w x v )  ->  u x
v ) )
6 2sp 1955 . . . . . . . 8  |-  ( A. w A. v ( ( u x w  /\  w x v )  ->  u x v )  ->  ( (
u x w  /\  w x v )  ->  u x v ) )
76sps 1954 . . . . . . 7  |-  ( A. u A. w A. v
( ( u x w  /\  w x v )  ->  u x v )  -> 
( ( u x w  /\  w x v )  ->  u x v ) )
83, 5, 73syl 18 . . . . . 6  |-  ( ph  ->  ( ( u x w  /\  w x v )  ->  u x v ) )
92, 8sylcom 30 . . . . 5  |-  ( ( ( ph  ->  u x w )  /\  ( ph  ->  w x
v ) )  -> 
( ph  ->  u x v ) )
109alanimi 1699 . . . 4  |-  ( ( A. x ( ph  ->  u x w )  /\  A. x (
ph  ->  w x v ) )  ->  A. x
( ph  ->  u x v ) )
11 opex 4681 . . . . . . 7  |-  <. u ,  w >.  e.  _V
1211elintab 4259 . . . . . 6  |-  ( <.
u ,  w >.  e. 
|^| { x  |  ph } 
<-> 
A. x ( ph  -> 
<. u ,  w >.  e.  x ) )
13 df-br 4419 . . . . . 6  |-  ( u
|^| { x  |  ph } w  <->  <. u ,  w >.  e.  |^| { x  | 
ph } )
14 df-br 4419 . . . . . . . 8  |-  ( u x w  <->  <. u ,  w >.  e.  x
)
1514imbi2i 318 . . . . . . 7  |-  ( (
ph  ->  u x w )  <->  ( ph  ->  <.
u ,  w >.  e.  x ) )
1615albii 1702 . . . . . 6  |-  ( A. x ( ph  ->  u x w )  <->  A. x
( ph  ->  <. u ,  w >.  e.  x
) )
1712, 13, 163bitr4i 285 . . . . 5  |-  ( u
|^| { x  |  ph } w  <->  A. x ( ph  ->  u x w ) )
18 opex 4681 . . . . . . 7  |-  <. w ,  v >.  e.  _V
1918elintab 4259 . . . . . 6  |-  ( <.
w ,  v >.  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  <. w ,  v >.  e.  x
) )
20 df-br 4419 . . . . . 6  |-  ( w
|^| { x  |  ph } v  <->  <. w ,  v >.  e.  |^| { x  |  ph } )
21 df-br 4419 . . . . . . . 8  |-  ( w x v  <->  <. w ,  v >.  e.  x
)
2221imbi2i 318 . . . . . . 7  |-  ( (
ph  ->  w x v )  <->  ( ph  ->  <.
w ,  v >.  e.  x ) )
2322albii 1702 . . . . . 6  |-  ( A. x ( ph  ->  w x v )  <->  A. x
( ph  ->  <. w ,  v >.  e.  x
) )
2419, 20, 233bitr4i 285 . . . . 5  |-  ( w
|^| { x  |  ph } v  <->  A. x
( ph  ->  w x v ) )
2517, 24anbi12i 708 . . . 4  |-  ( ( u |^| { x  |  ph } w  /\  w |^| { x  | 
ph } v )  <-> 
( A. x (
ph  ->  u x w )  /\  A. x
( ph  ->  w x v ) ) )
26 opex 4681 . . . . . 6  |-  <. u ,  v >.  e.  _V
2726elintab 4259 . . . . 5  |-  ( <.
u ,  v >.  e.  |^| { x  | 
ph }  <->  A. x
( ph  ->  <. u ,  v >.  e.  x
) )
28 df-br 4419 . . . . 5  |-  ( u
|^| { x  |  ph } v  <->  <. u ,  v >.  e.  |^| { x  |  ph } )
29 df-br 4419 . . . . . . 7  |-  ( u x v  <->  <. u ,  v >.  e.  x
)
3029imbi2i 318 . . . . . 6  |-  ( (
ph  ->  u x v )  <->  ( ph  ->  <.
u ,  v >.  e.  x ) )
3130albii 1702 . . . . 5  |-  ( A. x ( ph  ->  u x v )  <->  A. x
( ph  ->  <. u ,  v >.  e.  x
) )
3227, 28, 313bitr4i 285 . . . 4  |-  ( u
|^| { x  |  ph } v  <->  A. x
( ph  ->  u x v ) )
3310, 25, 323imtr4i 274 . . 3  |-  ( ( u |^| { x  |  ph } w  /\  w |^| { x  | 
ph } v )  ->  u |^| { x  |  ph } v )
3433gen2 1681 . 2  |-  A. w A. v ( ( u
|^| { x  |  ph } w  /\  w |^| { x  |  ph } v )  ->  u |^| { x  | 
ph } v )
351, 34mpgbir 1684 1  |-  ( |^| { x  |  ph }  o.  |^| { x  | 
ph } )  C_  |^|
{ x  |  ph }
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375   A.wal 1453    e. wcel 1898   {cab 2448    C_ wss 3416   <.cop 3986   |^|cint 4248   class class class wbr 4418    o. ccom 4860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-int 4249  df-br 4419  df-opab 4478  df-xp 4862  df-rel 4863  df-co 4865
This theorem is referenced by:  dfrtrcl5  36282
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