Theorem intabssd 37916

Description: When for each element y there is a subset A which may substituted for x such that y satisfying ch implies x satisfies ps then the intersection of all x that satisfy ps is a subclass the intersection of all y that satisfy ch. (Contributed by RP, 17-Oct-2020.)

Hypotheses

Ref	Expression
intabssd.ex	|- ( ph -> A e. V )
intabssd.sub	|- ( ( ph /\ x = A ) -> ( ch -> ps ) )
intabssd.ss	|- ( ph -> A C_ y )

Assertion

Ref	Expression
intabssd	|- ( ph -> |^| { x | ps } C_ |^| { y | ch } )

Distinct variable groups:   ch, x   ps, y   x, y, ph   x, A

Allowed substitution hints:   ps( x)   ch( y)   A( y)   V( x, y)

Proof of Theorem intabssd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intabssd.ex	. . . . 5 |- ( ph -> A e. V )
2		intabssd.sub	. . . . . 6 |- ( ( ph /\ x = A ) -> ( ch -> ps ) )
3		eleq2 2690	. . . . . . . 8 |- ( x = A -> ( z e. x <-> z e. A ) )
4	3	biimpd 219	. . . . . . 7 |- ( x = A -> ( z e. x -> z e. A ) )
5		intabssd.ss	. . . . . . . 8 |- ( ph -> A C_ y )
6	5	sseld 3602	. . . . . . 7 |- ( ph -> ( z e. A -> z e. y ) )
7	4, 6	sylan9r 690	. . . . . 6 |- ( ( ph /\ x = A ) -> ( z e. x -> z e. y ) )
8	2, 7	imim12d 81	. . . . 5 |- ( ( ph /\ x = A ) -> ( ( ps -> z e. x ) -> ( ch -> z e. y ) ) )
9	1, 8	spcimdv 3290	. . . 4 |- ( ph -> ( A. x ( ps -> z e. x ) -> ( ch -> z e. y ) ) )
10	9	alrimdv 1857	. . 3 |- ( ph -> ( A. x ( ps -> z e. x ) -> A. y ( ch -> z e. y ) ) )
11		vex 3203	. . . 4 |- z e. _V
12	11	elintab 4487	. . 3 |- ( z e. |^| { x | ps } <-> A. x ( ps -> z e. x ) )
13	11	elintab 4487	. . 3 |- ( z e. |^| { y | ch } <-> A. y ( ch -> z e. y ) )
14	10, 12, 13	3imtr4g 285	. 2 |- ( ph -> ( z e. |^| { x | ps } -> z e. |^| { y | ch } ) )
15	14	ssrdv 3609	1 |- ( ph -> |^| { x | ps } C_ |^| { y | ch } )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   C_ wss 3574   |^|cint 4475

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-int 4476

This theorem is referenced by:  clcnvlem  37930