Theorem riinm 3729

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 3129	. 2 |- ( A i^i |^|_ x e. X S ) = ( |^|_ x e. X S i^i A )
2		r19.2m 3309	. . . . 5 |- ( ( E. x x e. X /\ A. x e. X S C_ A ) -> E. x e. X S C_ A )
3	2	ancoms 255	. . . 4 |- ( ( A. x e. X S C_ A /\ E. x x e. X ) -> E. x e. X S C_ A )
4		iinss 3708	. . . 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 2931	. . 3 |- ( |^|_ x e. X S C_ A <-> ( |^|_ x e. X S i^i A ) = |^|_ x e. X S )
7	5, 6	sylib 127	. 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 2084	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 97   = wceq 1243   E.wex 1381   e. wcel 1393   A.wral 2306   E.wrex 2307   i^i cin 2916   C_ wss 2917   |^|_ciin 3658

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-iin 3660

This theorem is referenced by:  riinerm  6179