Theorem inintabss 37884

Description: Upper bound on intersection of class and the intersection of a class. (Contributed by RP, 13-Aug-2020.)

Assertion

Ref	Expression
inintabss	|- ( A i^i |^| { x | ph } ) C_ |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) }

Distinct variable groups:   ph, w   x, w, A

Allowed substitution hint:   ph( x)

Proof of Theorem inintabss

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1 6	. . . 4 |- ( u e. A -> ( E. x ph -> u e. A ) )
2	1	anim1i 592	. . 3 |- ( ( u e. A /\ A. x ( ph -> u e. x ) ) -> ( ( E. x ph -> u e. A ) /\ A. x ( ph -> u e. x ) ) )
3		elinintab 37881	. . 3 |- ( u e. ( A i^i |^| { x | ph } ) <-> ( u e. A /\ A. x ( ph -> u e. x ) ) )
4		vex 3203	. . . 4 |- u e. _V
5		elinintrab 37883	. . . 4 |- ( u e. _V -> ( u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) } <-> ( ( E. x ph -> u e. A ) /\ A. x ( ph -> u e. x ) ) ) )
6	4, 5	ax-mp 5	. . 3 |- ( u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) } <-> ( ( E. x ph -> u e. A ) /\ A. x ( ph -> u e. x ) ) )
7	2, 3, 6	3imtr4i 281	. 2 |- ( u e. ( A i^i |^| { x | ph } ) -> u e. |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) } )
8	7	ssriv 3607	1 |- ( A i^i |^| { x | ph } ) C_ |^| { w e. ~P A | E. x ( w = ( A i^i x ) /\ ph ) }

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

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  ax-sep 4781

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-int 4476

This theorem is referenced by: (None)