Theorem riiner 7820

Description: The relative intersection of a family of equivalence relations is an equivalence relation. (Contributed by Mario Carneiro, 27-Sep-2015.)

Assertion

Ref	Expression
riiner	|- ( A. x e. A R Er B -> ( ( B X. B ) i^i |^|_ x e. A R ) Er B )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   R( x)

Proof of Theorem riiner

Step	Hyp	Ref	Expression
1		xpider 7818	. . 3 |- ( B X. B ) Er B
2		riin0 4594	. . . . 5 |- ( A = (/) -> ( ( B X. B ) i^i |^|_ x e. A R ) = ( B X. B ) )
3	2	adantl 482	. . . 4 |- ( ( A. x e. A R Er B /\ A = (/) ) -> ( ( B X. B ) i^i |^|_ x e. A R ) = ( B X. B ) )
4		ereq1 7749	. . . 4 |- ( ( ( B X. B ) i^i |^|_ x e. A R ) = ( B X. B ) -> ( ( ( B X. B ) i^i |^|_ x e. A R ) Er B <-> ( B X. B ) Er B ) )
5	3, 4	syl 17	. . 3 |- ( ( A. x e. A R Er B /\ A = (/) ) -> ( ( ( B X. B ) i^i |^|_ x e. A R ) Er B <-> ( B X. B ) Er B ) )
6	1, 5	mpbiri 248	. 2 |- ( ( A. x e. A R Er B /\ A = (/) ) -> ( ( B X. B ) i^i |^|_ x e. A R ) Er B )
7		iiner 7819	. . . 4 |- ( ( A =/= (/) /\ A. x e. A R Er B ) -> |^|_ x e. A R Er B )
8	7	ancoms 469	. . 3 |- ( ( A. x e. A R Er B /\ A =/= (/) ) -> |^|_ x e. A R Er B )
9		erssxp 7765	. . . . . 6 |- ( R Er B -> R C_ ( B X. B ) )
10	9	ralimi 2952	. . . . 5 |- ( A. x e. A R Er B -> A. x e. A R C_ ( B X. B ) )
11		riinn0 4595	. . . . 5 |- ( ( A. x e. A R C_ ( B X. B ) /\ A =/= (/) ) -> ( ( B X. B ) i^i |^|_ x e. A R ) = |^|_ x e. A R )
12	10, 11	sylan 488	. . . 4 |- ( ( A. x e. A R Er B /\ A =/= (/) ) -> ( ( B X. B ) i^i |^|_ x e. A R ) = |^|_ x e. A R )
13		ereq1 7749	. . . 4 |- ( ( ( B X. B ) i^i |^|_ x e. A R ) = |^|_ x e. A R -> ( ( ( B X. B ) i^i |^|_ x e. A R ) Er B <-> |^|_ x e. A R Er B ) )
14	12, 13	syl 17	. . 3 |- ( ( A. x e. A R Er B /\ A =/= (/) ) -> ( ( ( B X. B ) i^i |^|_ x e. A R ) Er B <-> |^|_ x e. A R Er B ) )
15	8, 14	mpbird 247	. 2 |- ( ( A. x e. A R Er B /\ A =/= (/) ) -> ( ( B X. B ) i^i |^|_ x e. A R ) Er B )
16	6, 15	pm2.61dane 2881	1 |- ( A. x e. A R Er B -> ( ( B X. B ) i^i |^|_ x e. A R ) Er B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   =/= wne 2794   A.wral 2912   i^i cin 3573   C_ wss 3574   (/)c0 3915   |^|_ciin 4521   X. cxp 5112   Er wer 7739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iin 4523  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-er 7742

This theorem is referenced by: (None)