Theorem riinm 4038

Description: Relative intersection of an inhabited family. (Contributed by Jim Kingdon, 19-Aug-2018.)

Assertion

Ref	Expression
riinm	|- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> ( A i^i |^|_ x e. X S ) = |^|_ x e. X S )

Distinct variable groups:   x, A   x, X

Allowed substitution hint:   S( x)

Proof of Theorem riinm

Step	Hyp	Ref	Expression
1		incom 3396	. 2 |- ( A i^i |^|_ x e. X S ) = ( |^|_ x e. X S i^i A )
2		r19.2m 3578	. . . . 5 |- ( ( E. x x e. X /\ A. x e. X S C_ A ) -> E. x e. X S C_ A )
3	2	ancoms 268	. . . 4 |- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> E. x e. X S C_ A )
4		iinss 4017	. . . 4 |- ( E. x e. X S C_ A -> |^|_ x e. X S C_ A )
5	3, 4	syl 14	. . 3 |- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> |^|_ x e. X S C_ A )
6		df-ss 3210	. . 3 |- ( |^|_ x e. X S C_ A <-> ( |^|_ x e. X S i^i A ) = |^|_ x e. X S )
7	5, 6	sylib 122	. 2 |- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> ( |^|_ x e. X S i^i A ) = |^|_ x e. X S )
8	1, 7	eqtrid 2274	1 |- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> ( A i^i |^|_ x e. X S ) = |^|_ x e. X S )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   E.wex 1538   e. wcel 2200   A.wral 2508   E.wrex 2509   i^i cin 3196   C_ wss 3197   |^|_ciin 3966

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-in 3203  df-ss 3210  df-iin 3968

This theorem is referenced by:  riinerm  6755