Theorem rintn0 4370

Description: Relative intersection of a nonempty set. (Contributed by Stefan O'Rear, 3-Apr-2015.) (Revised by Mario Carneiro, 5-Jun-2015.)

Assertion

Ref	Expression
rintn0	|- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X )

Proof of Theorem rintn0

Step	Hyp	Ref	Expression
1		incom 3652	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2 4262	. . . 4 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ U. ~P A )
3		ssid 3484	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 4365	. . . . 5 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
5	3, 4	mpbi 208	. . . 4 |- U. ~P A C_ A
6	2, 5	syl6ss 3477	. . 3 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ A )
7		df-ss 3451	. . 3 |- ( |^| X C_ A <-> ( |^| X i^i A ) = |^| X )
8	6, 7	sylib 196	. 2 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( |^| X i^i A ) = |^| X )
9	1, 8	syl5eq 2507	1 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   =/= wne 2648   i^i cin 3436   C_ wss 3437   (/)c0 3746   ~Pcpw 3969   U.cuni 4200   |^|cint 4237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747  df-pw 3971  df-uni 4201  df-int 4238

This theorem is referenced by:  mrerintcl  14655  ismred2  14661