Theorem cnvcnvintabd 37906

Description: Value of the relationship content of the intersection of a class. (Contributed by RP, 20-Aug-2020.)

Hypothesis

Ref	Expression
cnvcnvintabd.x	|- ( ph -> E. x ps )

Assertion

Ref	Expression
cnvcnvintabd	|- ( ph -> `' `' |^| { x | ps } = |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' `' x /\ ps ) } )

Distinct variable groups:   ps, w   x, w

Allowed substitution hints:   ph( x, w)   ps( x)

Proof of Theorem cnvcnvintabd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvcnv 5586	. . . . . . . . . 10 |- `' `' x = ( x i^i ( _V X. _V ) )
2	1	eleq2i 2693	. . . . . . . . 9 |- ( y e. `' `' x <-> y e. ( x i^i ( _V X. _V ) ) )
3		elin 3796	. . . . . . . . . 10 |- ( y e. ( x i^i ( _V X. _V ) ) <-> ( y e. x /\ y e. ( _V X. _V ) ) )
4	3	rbaib 947	. . . . . . . . 9 |- ( y e. ( _V X. _V ) -> ( y e. ( x i^i ( _V X. _V ) ) <-> y e. x ) )
5	2, 4	syl5bb 272	. . . . . . . 8 |- ( y e. ( _V X. _V ) -> ( y e. `' `' x <-> y e. x ) )
6	5	bicomd 213	. . . . . . 7 |- ( y e. ( _V X. _V ) -> ( y e. x <-> y e. `' `' x ) )
7	6	imbi2d 330	. . . . . 6 |- ( y e. ( _V X. _V ) -> ( ( ps -> y e. x ) <-> ( ps -> y e. `' `' x ) ) )
8	7	albidv 1849	. . . . 5 |- ( y e. ( _V X. _V ) -> ( A. x ( ps -> y e. x ) <-> A. x ( ps -> y e. `' `' x ) ) )
9	8	pm5.32i 669	. . . 4 |- ( ( y e. ( _V X. _V ) /\ A. x ( ps -> y e. x ) ) <-> ( y e. ( _V X. _V ) /\ A. x ( ps -> y e. `' `' x ) ) )
10		cnvcnvintabd.x	. . . . . . 7 |- ( ph -> E. x ps )
11		pm5.5 351	. . . . . . 7 |- ( E. x ps -> ( ( E. x ps -> y e. ( _V X. _V ) ) <-> y e. ( _V X. _V ) ) )
12	10, 11	syl 17	. . . . . 6 |- ( ph -> ( ( E. x ps -> y e. ( _V X. _V ) ) <-> y e. ( _V X. _V ) ) )
13	12	bicomd 213	. . . . 5 |- ( ph -> ( y e. ( _V X. _V ) <-> ( E. x ps -> y e. ( _V X. _V ) ) ) )
14	13	anbi1d 741	. . . 4 |- ( ph -> ( ( y e. ( _V X. _V ) /\ A. x ( ps -> y e. `' `' x ) ) <-> ( ( E. x ps -> y e. ( _V X. _V ) ) /\ A. x ( ps -> y e. `' `' x ) ) ) )
15	9, 14	syl5bb 272	. . 3 |- ( ph -> ( ( y e. ( _V X. _V ) /\ A. x ( ps -> y e. x ) ) <-> ( ( E. x ps -> y e. ( _V X. _V ) ) /\ A. x ( ps -> y e. `' `' x ) ) ) )
16		elcnvcnvintab 37888	. . 3 |- ( y e. `' `' |^| { x | ps } <-> ( y e. ( _V X. _V ) /\ A. x ( ps -> y e. x ) ) )
17		vex 3203	. . . 4 |- y e. _V
18		vex 3203	. . . . . 6 |- x e. _V
19		cnvexg 7112	. . . . . 6 |- ( x e. _V -> `' x e. _V )
20		cnvexg 7112	. . . . . 6 |- ( `' x e. _V -> `' `' x e. _V )
21	18, 19, 20	mp2b 10	. . . . 5 |- `' `' x e. _V
22		relcnv 5503	. . . . . 6 |- Rel `' `' x
23		df-rel 5121	. . . . . 6 |- ( Rel `' `' x <-> `' `' x C_ ( _V X. _V ) )
24	22, 23	mpbi 220	. . . . 5 |- `' `' x C_ ( _V X. _V )
25	21, 24	elmapintrab 37882	. . . 4 |- ( y e. _V -> ( y e. |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' `' x /\ ps ) } <-> ( ( E. x ps -> y e. ( _V X. _V ) ) /\ A. x ( ps -> y e. `' `' x ) ) ) )
26	17, 25	ax-mp 5	. . 3 |- ( y e. |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' `' x /\ ps ) } <-> ( ( E. x ps -> y e. ( _V X. _V ) ) /\ A. x ( ps -> y e. `' `' x ) ) )
27	15, 16, 26	3bitr4g 303	. 2 |- ( ph -> ( y e. `' `' |^| { x | ps } <-> y e. |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' `' x /\ ps ) } ) )
28	27	eqrdv 2620	1 |- ( ph -> `' `' |^| { x | ps } = |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' `' x /\ ps ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~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-pow 4843  ax-pr 4906  ax-un 6949

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-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-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by: (None)