Theorem bj-0int 33055

Description: If A is a collection of subsets of X, like a topology, two equivalent ways to say that arbitrary intersections of elements of A relative to X belong to some class B (in typical applications, A itself). (Contributed by BJ, 7-Dec-2021.)

Assertion

Ref	Expression
bj-0int	|- ( A C_ ~P X -> ( ( X e. B /\ A. x e. ( ~P A \ { (/) } ) |^| x e. B ) <-> A. x e. ~P A ( X i^i |^| x ) e. B ) )

Distinct variable groups:   x, A   x, B   x, X

Proof of Theorem bj-0int

Step	Hyp	Ref	Expression
1		ssv 3625	. . . . . . . . 9 |- X C_ _V
2		int0 4490	. . . . . . . . 9 |- |^| (/) = _V
3	1, 2	sseqtr4i 3638	. . . . . . . 8 |- X C_ |^| (/)
4		df-ss 3588	. . . . . . . 8 |- ( X C_ |^| (/) <-> ( X i^i |^| (/) ) = X )
5	3, 4	mpbi 220	. . . . . . 7 |- ( X i^i |^| (/) ) = X
6	5	eqcomi 2631	. . . . . 6 |- X = ( X i^i |^| (/) )
7	6	eleq1i 2692	. . . . 5 |- ( X e. B <-> ( X i^i |^| (/) ) e. B )
8	7	a1i 11	. . . 4 |- ( A C_ ~P X -> ( X e. B <-> ( X i^i |^| (/) ) e. B ) )
9		eldifsn 4317	. . . . . . . 8 |- ( x e. ( ~P A \ { (/) } ) <-> ( x e. ~P A /\ x =/= (/) ) )
10		sstr2 3610	. . . . . . . . . . 11 |- ( x C_ A -> ( A C_ ~P X -> x C_ ~P X ) )
11		bj-intss 33053	. . . . . . . . . . 11 |- ( x C_ ~P X -> ( x =/= (/) -> |^| x C_ X ) )
12	10, 11	syl6 35	. . . . . . . . . 10 |- ( x C_ A -> ( A C_ ~P X -> ( x =/= (/) -> |^| x C_ X ) ) )
13		elpwi 4168	. . . . . . . . . 10 |- ( x e. ~P A -> x C_ A )
14	12, 13	syl11 33	. . . . . . . . 9 |- ( A C_ ~P X -> ( x e. ~P A -> ( x =/= (/) -> |^| x C_ X ) ) )
15	14	impd 447	. . . . . . . 8 |- ( A C_ ~P X -> ( ( x e. ~P A /\ x =/= (/) ) -> |^| x C_ X ) )
16	9, 15	syl5bi 232	. . . . . . 7 |- ( A C_ ~P X -> ( x e. ( ~P A \ { (/) } ) -> |^| x C_ X ) )
17		df-ss 3588	. . . . . . . . 9 |- ( |^| x C_ X <-> ( |^| x i^i X ) = |^| x )
18		incom 3805	. . . . . . . . . . 11 |- ( |^| x i^i X ) = ( X i^i |^| x )
19	18	eqeq1i 2627	. . . . . . . . . 10 |- ( ( |^| x i^i X ) = |^| x <-> ( X i^i |^| x ) = |^| x )
20		eqcom 2629	. . . . . . . . . 10 |- ( ( X i^i |^| x ) = |^| x <-> |^| x = ( X i^i |^| x ) )
21	19, 20	sylbb 209	. . . . . . . . 9 |- ( ( |^| x i^i X ) = |^| x -> |^| x = ( X i^i |^| x ) )
22	17, 21	sylbi 207	. . . . . . . 8 |- ( |^| x C_ X -> |^| x = ( X i^i |^| x ) )
23		eleq1 2689	. . . . . . . . 9 |- ( |^| x = ( X i^i |^| x ) -> ( |^| x e. B <-> ( X i^i |^| x ) e. B ) )
24	23	a1i 11	. . . . . . . 8 |- ( A C_ ~P X -> ( |^| x = ( X i^i |^| x ) -> ( |^| x e. B <-> ( X i^i |^| x ) e. B ) ) )
25	22, 24	syl5 34	. . . . . . 7 |- ( A C_ ~P X -> ( |^| x C_ X -> ( |^| x e. B <-> ( X i^i |^| x ) e. B ) ) )
26	16, 25	syld 47	. . . . . 6 |- ( A C_ ~P X -> ( x e. ( ~P A \ { (/) } ) -> ( |^| x e. B <-> ( X i^i |^| x ) e. B ) ) )
27	26	ralrimiv 2965	. . . . 5 |- ( A C_ ~P X -> A. x e. ( ~P A \ { (/) } ) ( |^| x e. B <-> ( X i^i |^| x ) e. B ) )
28		ralbi 3068	. . . . 5 |- ( A. x e. ( ~P A \ { (/) } ) ( |^| x e. B <-> ( X i^i |^| x ) e. B ) -> ( A. x e. ( ~P A \ { (/) } ) |^| x e. B <-> A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B ) )
29	27, 28	syl 17	. . . 4 |- ( A C_ ~P X -> ( A. x e. ( ~P A \ { (/) } ) |^| x e. B <-> A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B ) )
30	8, 29	anbi12d 747	. . 3 |- ( A C_ ~P X -> ( ( X e. B /\ A. x e. ( ~P A \ { (/) } ) |^| x e. B ) <-> ( ( X i^i |^| (/) ) e. B /\ A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B ) ) )
31		ancom 466	. . 3 |- ( ( ( X i^i |^| (/) ) e. B /\ A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B ) <-> ( A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B /\ ( X i^i |^| (/) ) e. B ) )
32	30, 31	syl6bb 276	. 2 |- ( A C_ ~P X -> ( ( X e. B /\ A. x e. ( ~P A \ { (/) } ) |^| x e. B ) <-> ( A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B /\ ( X i^i |^| (/) ) e. B ) ) )
33		0elpw 4834	. . 3 |- (/) e. ~P A
34		inteq 4478	. . . . 5 |- ( x = (/) -> |^| x = |^| (/) )
35		ineq2 3808	. . . . 5 |- ( |^| x = |^| (/) -> ( X i^i |^| x ) = ( X i^i |^| (/) ) )
36		eleq1 2689	. . . . 5 |- ( ( X i^i |^| x ) = ( X i^i |^| (/) ) -> ( ( X i^i |^| x ) e. B <-> ( X i^i |^| (/) ) e. B ) )
37	34, 35, 36	3syl 18	. . . 4 |- ( x = (/) -> ( ( X i^i |^| x ) e. B <-> ( X i^i |^| (/) ) e. B ) )
38	37	bj-raldifsn 33054	. . 3 |- ( (/) e. ~P A -> ( A. x e. ~P A ( X i^i |^| x ) e. B <-> ( A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B /\ ( X i^i |^| (/) ) e. B ) ) )
39	33, 38	ax-mp 5	. 2 |- ( A. x e. ~P A ( X i^i |^| x ) e. B <-> ( A. x e. ( ~P A \ { (/) } ) ( X i^i |^| x ) e. B /\ ( X i^i |^| (/) ) e. B ) )
40	32, 39	syl6bbr 278	1 |- ( A C_ ~P X -> ( ( X e. B /\ A. x e. ( ~P A \ { (/) } ) |^| x e. B ) <-> A. x e. ~P A ( X i^i |^| x ) e. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|cint 4475

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-nul 4789

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-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-pw 4160  df-sn 4178  df-uni 4437  df-int 4476

This theorem is referenced by: (None)