Theorem rintn0 4371

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 3624	. 2 |- ( A i^i |^| X ) = ( |^| X i^i A )
2		intssuni2 4259	. . . 4 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ U. ~P A )
3		ssid 3450	. . . . 5 |- ~P A C_ ~P A
4		sspwuni 4366	. . . . 5 |- ( ~P A C_ ~P A <-> U. ~P A C_ A )
5	3, 4	mpbi 212	. . . 4 |- U. ~P A C_ A
6	2, 5	syl6ss 3443	. . 3 |- ( ( X C_ ~P A /\ X =/= (/) ) -> |^| X C_ A )
7		df-ss 3417	. . 3 |- ( |^| X C_ A <-> ( |^| X i^i A ) = |^| X )
8	6, 7	sylib 200	. 2 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( |^| X i^i A ) = |^| X )
9	1, 8	syl5eq 2496	1 |- ( ( X C_ ~P A /\ X =/= (/) ) -> ( A i^i |^| X ) = |^| X )

Syntax hints:   -> wi 4   /\ wa 371   = wceq 1443   =/= wne 2621   i^i cin 3402   C_ wss 3403   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   |^|cint 4233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-v 3046  df-dif 3406  df-in 3410  df-ss 3417  df-nul 3731  df-pw 3952  df-uni 4198  df-int 4234

This theorem is referenced by:  mrerintcl  15496  ismred2  15502