Theorem riinm 3750

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 3158	. 2 |- ( A i^i |^|_ x e. X S ) = ( |^|_ x e. X S i^i A )
2		r19.2m 3329	. . . . 5 |- ( ( E. x x e. X /\ A. x e. X S C_ A ) -> E. x e. X S C_ A )
3	2	ancoms 264	. . . 4 |- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> E. x e. X S C_ A )
4		iinss 3729	. . . 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 2986	. . 3 |- ( |^|_ x e. X S C_ A <-> ( |^|_ x e. X S i^i A ) = |^|_ x e. X S )
7	5, 6	sylib 120	. 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 2125	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 102   = wceq 1284   E.wex 1421   e. wcel 1433   A.wral 2348   E.wrex 2349   i^i cin 2972   C_ wss 2973   |^|_ciin 3679

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-in 2979  df-ss 2986  df-iin 3681

This theorem is referenced by:  riinerm  6202