Theorem relintab 37889

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

Step	Hyp	Ref	Expression
1		cnvcnv 5586	. . 3 |- `' `' |^| { x | ph } = ( |^| { x | ph } i^i ( _V X. _V ) )
2		incom 3805	. . 3 |- ( |^| { x | ph } i^i ( _V X. _V ) ) = ( ( _V X. _V ) i^i |^| { x | ph } )
3	1, 2	eqtri 2644	. 2 |- `' `' |^| { x | ph } = ( ( _V X. _V ) i^i |^| { x | ph } )
4		dfrel2 5583	. . 3 |- ( Rel |^| { x | ph } <-> `' `' |^| { x | ph } = |^| { x | ph } )
5	4	biimpi 206	. 2 |- ( Rel |^| { x | ph } -> `' `' |^| { x | ph } = |^| { x | ph } )
6		relintabex 37887	. . . 4 |- ( Rel |^| { x | ph } -> E. x ph )
7	6	xpinintabd 37886	. . 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 3805	. . . . . . . . . 10 |- ( ( _V X. _V ) i^i x ) = ( x i^i ( _V X. _V ) )
9		cnvcnv 5586	. . . . . . . . . 10 |- `' `' x = ( x i^i ( _V X. _V ) )
10	8, 9	eqtr4i 2647	. . . . . . . . 9 |- ( ( _V X. _V ) i^i x ) = `' `' x
11	10	eqeq2i 2634	. . . . . . . 8 |- ( w = ( ( _V X. _V ) i^i x ) <-> w = `' `' x )
12	11	anbi1i 731	. . . . . . 7 |- ( ( w = ( ( _V X. _V ) i^i x ) /\ ph ) <-> ( w = `' `' x /\ ph ) )
13	12	exbii 1774	. . . . . 6 |- ( E. x ( w = ( ( _V X. _V ) i^i x ) /\ ph ) <-> E. x ( w = `' `' x /\ ph ) )
14	13	a1i 11	. . . . 5 |- ( w e. ~P ( _V X. _V ) -> ( E. x ( w = ( ( _V X. _V ) i^i x ) /\ ph ) <-> E. x ( w = `' `' x /\ ph ) ) )
15	14	rabbiia 3185	. . . 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 ) }
16	15	inteqi 4479	. . 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 ) }
17	7, 16	syl6eq 2672	. 2 |- ( Rel |^| { x | ph } -> ( ( _V X. _V ) i^i |^| { x | ph } ) = |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' `' x /\ ph ) } )
18	3, 5, 17	3eqtr3a 2680	1 |- ( Rel |^| { x | ph } -> |^| { x | ph } = |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' `' x /\ ph ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   i^i cin 3573   ~Pcpw 4158   |^|cint 4475   X. cxp 5112   `'ccnv 5113   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122

This theorem is referenced by: (None)