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Theorem relintab 36260
Description: Value of the intersection of a class when it is a relation. (Contributed by RP, 12-Aug-2020.)
Assertion
Ref Expression
relintab  |-  ( Rel  |^| { x  |  ph }  ->  |^| { x  | 
ph }  =  |^| { w  e.  ~P ( _V  X.  _V )  |  E. x ( w  =  `' `' x  /\  ph ) } )
Distinct variable groups:    ph, w    x, w
Allowed substitution hint:    ph( x)

Proof of Theorem relintab
StepHypRef Expression
1 cnvcnv 5295 . . 3  |-  `' `' |^| { x  |  ph }  =  ( |^| { x  |  ph }  i^i  ( _V  X.  _V ) )
2 incom 3616 . . 3  |-  ( |^| { x  |  ph }  i^i  ( _V  X.  _V ) )  =  ( ( _V  X.  _V )  i^i  |^| { x  | 
ph } )
31, 2eqtri 2493 . 2  |-  `' `' |^| { x  |  ph }  =  ( ( _V  X.  _V )  i^i  |^| { x  |  ph } )
4 dfrel2 5292 . . 3  |-  ( Rel  |^| { x  |  ph } 
<->  `' `' |^| { x  | 
ph }  =  |^| { x  |  ph }
)
54biimpi 199 . 2  |-  ( Rel  |^| { x  |  ph }  ->  `' `' |^| { x  |  ph }  =  |^| { x  | 
ph } )
6 relintabex 36258 . . . 4  |-  ( Rel  |^| { x  |  ph }  ->  E. x ph )
76xpinintabd 36257 . . 3  |-  ( Rel  |^| { x  |  ph }  ->  ( ( _V 
X.  _V )  i^i  |^| { x  |  ph }
)  =  |^| { w  e.  ~P ( _V  X.  _V )  |  E. x ( w  =  ( ( _V  X.  _V )  i^i  x
)  /\  ph ) } )
8 incom 3616 . . . . . . . . . 10  |-  ( ( _V  X.  _V )  i^i  x )  =  ( x  i^i  ( _V 
X.  _V ) )
9 cnvcnv 5295 . . . . . . . . . 10  |-  `' `' x  =  ( x  i^i  ( _V  X.  _V ) )
108, 9eqtr4i 2496 . . . . . . . . 9  |-  ( ( _V  X.  _V )  i^i  x )  =  `' `' x
1110eqeq2i 2483 . . . . . . . 8  |-  ( w  =  ( ( _V 
X.  _V )  i^i  x
)  <->  w  =  `' `' x )
1211anbi1i 709 . . . . . . 7  |-  ( ( w  =  ( ( _V  X.  _V )  i^i  x )  /\  ph ) 
<->  ( w  =  `' `' x  /\  ph )
)
1312exbii 1726 . . . . . 6  |-  ( E. x ( w  =  ( ( _V  X.  _V )  i^i  x
)  /\  ph )  <->  E. x
( w  =  `' `' x  /\  ph )
)
1413a1i 11 . . . . 5  |-  ( w  e.  ~P ( _V 
X.  _V )  ->  ( E. x ( w  =  ( ( _V  X.  _V )  i^i  x
)  /\  ph )  <->  E. x
( w  =  `' `' x  /\  ph )
) )
1514rabbiia 3019 . . . 4  |-  { w  e.  ~P ( _V  X.  _V )  |  E. x ( w  =  ( ( _V  X.  _V )  i^i  x
)  /\  ph ) }  =  { w  e. 
~P ( _V  X.  _V )  |  E. x ( w  =  `' `' x  /\  ph ) }
1615inteqi 4230 . . 3  |-  |^| { w  e.  ~P ( _V  X.  _V )  |  E. x ( w  =  ( ( _V  X.  _V )  i^i  x
)  /\  ph ) }  =  |^| { w  e.  ~P ( _V  X.  _V )  |  E. x ( w  =  `' `' x  /\  ph ) }
177, 16syl6eq 2521 . 2  |-  ( Rel  |^| { x  |  ph }  ->  ( ( _V 
X.  _V )  i^i  |^| { x  |  ph }
)  =  |^| { w  e.  ~P ( _V  X.  _V )  |  E. x ( w  =  `' `' x  /\  ph ) } )
183, 5, 173eqtr3a 2529 1  |-  ( Rel  |^| { x  |  ph }  ->  |^| { x  | 
ph }  =  |^| { w  e.  ~P ( _V  X.  _V )  |  E. x ( w  =  `' `' x  /\  ph ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   {crab 2760   _Vcvv 3031    i^i cin 3389   ~Pcpw 3942   |^|cint 4226    X. cxp 4837   `'ccnv 4838   Rel wrel 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-int 4227  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847
This theorem is referenced by: (None)
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