Theorem cnvintabd 37909

Description: Value of the converse of the intersection of a non-empty class. (Contributed by RP, 20-Aug-2020.)

Hypothesis

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

Assertion

Ref	Expression
cnvintabd	|- ( 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 cnvintabd

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnvintabd.x	. . . . . 6 |- ( ph -> E. x ps )
2		pm5.5 351	. . . . . 6 |- ( E. x ps -> ( ( E. x ps -> y e. ( _V X. _V ) ) <-> y e. ( _V X. _V ) ) )
3	1, 2	syl 17	. . . . 5 |- ( ph -> ( ( E. x ps -> y e. ( _V X. _V ) ) <-> y e. ( _V X. _V ) ) )
4	3	bicomd 213	. . . 4 |- ( ph -> ( y e. ( _V X. _V ) <-> ( E. x ps -> y e. ( _V X. _V ) ) ) )
5	4	anbi1d 741	. . 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 ) ) ) )
6		elcnvintab 37908	. . 3 |- ( y e. `' |^| { x | ps } <-> ( y e. ( _V X. _V ) /\ A. x ( ps -> y e. `' x ) ) )
7		vex 3203	. . . 4 |- y e. _V
8		vex 3203	. . . . . 6 |- x e. _V
9	8	cnvex 7113	. . . . 5 |- `' x e. _V
10		relcnv 5503	. . . . . 6 |- Rel `' x
11		df-rel 5121	. . . . . 6 |- ( Rel `' x <-> `' x C_ ( _V X. _V ) )
12	10, 11	mpbi 220	. . . . 5 |- `' x C_ ( _V X. _V )
13	9, 12	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 ) ) ) )
14	7, 13	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 ) ) )
15	5, 6, 14	3bitr4g 303	. 2 |- ( ph -> ( y e. `' |^| { x | ps } <-> y e. |^| { w e. ~P ( _V X. _V ) | E. x ( w = `' x /\ ps ) } ) )
16	15	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   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-sbc 3436  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-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  clcnvlem  37930