Theorem riinm 3880

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 3263	. 2 |- ( A i^i |^|_ x e. X S ) = ( |^|_ x e. X S i^i A )
2		r19.2m 3444	. . . . 5 |- ( ( E. x x e. X /\ A. x e. X S C_ A ) -> E. x e. X S C_ A )
3	2	ancoms 266	. . . 4 |- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> E. x e. X S C_ A )
4		iinss 3859	. . . 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 3079	. . 3 |- ( |^|_ x e. X S C_ A <-> ( |^|_ x e. X S i^i A ) = |^|_ x e. X S )
7	5, 6	sylib 121	. 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	syl5eq 2182	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 103   = wceq 1331   E.wex 1468   e. wcel 1480   A.wral 2414   E.wrex 2415   i^i cin 3065   C_ wss 3066   |^|_ciin 3809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-in 3072  df-ss 3079  df-iin 3811

This theorem is referenced by:  riinerm  6495