Theorem xpinintabd 37886

Description: Value of the intersection of cross-product with the intersection of a non-empty class. (Contributed by RP, 12-Aug-2020.)

Hypothesis

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

Assertion

Ref	Expression
xpinintabd	|- ( ph -> ( ( A X. B ) i^i |^| { x | ps } ) = |^| { w e. ~P ( A X. B ) | E. x ( w = ( ( A X. B ) i^i x ) /\ ps ) } )

Distinct variable groups:   ps, w   x, w, A   w, B, x

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

Proof of Theorem xpinintabd

Step	Hyp	Ref	Expression
1		xpinintabd.x	. 2 |- ( ph -> E. x ps )
2	1	inintabd 37885	1 |- ( ph -> ( ( A X. B ) i^i |^| { x | ps } ) = |^| { w e. ~P ( A X. B ) | E. x ( w = ( ( A X. B ) i^i x ) /\ ps ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   {cab 2608   {crab 2916   i^i cin 3573   ~Pcpw 4158   |^|cint 4475   X. cxp 5112

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:  relintab  37889