Theorem ispisys2 30216

Description: The property of being a pi-system, expanded version. Pi-systems are closed under finite intersections. (Contributed by Thierry Arnoux, 13-Jun-2020.)

Hypothesis

Ref	Expression
ispisys.p	|- P = { s e. ~P ~P O | ( fi ` s ) C_ s }

Assertion

Ref	Expression
ispisys2	|- ( S e. P <-> ( S e. ~P ~P O /\ A. x e. ( ( ~P S i^i Fin ) \ { (/) } ) |^| x e. S ) )

Distinct variable groups:   O, s, x   S, s, x

Allowed substitution hints:   P( x, s)

Proof of Theorem ispisys2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ispisys.p	. . 3 |- P = { s e. ~P ~P O | ( fi ` s ) C_ s }
2	1	ispisys 30215	. 2 |- ( S e. P <-> ( S e. ~P ~P O /\ ( fi ` S ) C_ S ) )
3		dfss3 3592	. . . 4 |- ( ( fi ` S ) C_ S <-> A. y e. ( fi ` S ) y e. S )
4		elex 3212	. . . . . . 7 |- ( S e. ~P ~P O -> S e. _V )
5	4	adantr 481	. . . . . 6 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> S e. _V )
6		simpr 477	. . . . . . . . . 10 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> x e. ( ( ~P S i^i Fin ) \ { (/) } ) )
7		eldifsn 4317	. . . . . . . . . 10 |- ( x e. ( ( ~P S i^i Fin ) \ { (/) } ) <-> ( x e. ( ~P S i^i Fin ) /\ x =/= (/) ) )
8	6, 7	sylib 208	. . . . . . . . 9 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> ( x e. ( ~P S i^i Fin ) /\ x =/= (/) ) )
9	8	simpld 475	. . . . . . . 8 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> x e. ( ~P S i^i Fin ) )
10	9	elin1d 3802	. . . . . . 7 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> x e. ~P S )
11	10	elpwid 4170	. . . . . 6 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> x C_ S )
12	8	simprd 479	. . . . . 6 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> x =/= (/) )
13	9	elin2d 3803	. . . . . 6 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> x e. Fin )
14		elfir 8321	. . . . . 6 |- ( ( S e. _V /\ ( x C_ S /\ x =/= (/) /\ x e. Fin ) ) -> |^| x e. ( fi ` S ) )
15	5, 11, 12, 13, 14	syl13anc 1328	. . . . 5 |- ( ( S e. ~P ~P O /\ x e. ( ( ~P S i^i Fin ) \ { (/) } ) ) -> |^| x e. ( fi ` S ) )
16		elfi2 8320	. . . . . 6 |- ( S e. ~P ~P O -> ( y e. ( fi ` S ) <-> E. x e. ( ( ~P S i^i Fin ) \ { (/) } ) y = |^| x ) )
17	16	biimpa 501	. . . . 5 |- ( ( S e. ~P ~P O /\ y e. ( fi ` S ) ) -> E. x e. ( ( ~P S i^i Fin ) \ { (/) } ) y = |^| x )
18		simpr 477	. . . . . 6 |- ( ( S e. ~P ~P O /\ y = |^| x ) -> y = |^| x )
19	18	eleq1d 2686	. . . . 5 |- ( ( S e. ~P ~P O /\ y = |^| x ) -> ( y e. S <-> |^| x e. S ) )
20	15, 17, 19	ralxfrd 4879	. . . 4 |- ( S e. ~P ~P O -> ( A. y e. ( fi ` S ) y e. S <-> A. x e. ( ( ~P S i^i Fin ) \ { (/) } ) |^| x e. S ) )
21	3, 20	syl5bb 272	. . 3 |- ( S e. ~P ~P O -> ( ( fi ` S ) C_ S <-> A. x e. ( ( ~P S i^i Fin ) \ { (/) } ) |^| x e. S ) )
22	21	pm5.32i 669	. 2 |- ( ( S e. ~P ~P O /\ ( fi ` S ) C_ S ) <-> ( S e. ~P ~P O /\ A. x e. ( ( ~P S i^i Fin ) \ { (/) } ) |^| x e. S ) )
23	2, 22	bitri 264	1 |- ( S e. P <-> ( S e. ~P ~P O /\ A. x e. ( ( ~P S i^i Fin ) \ { (/) } ) |^| x e. S ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   |^|cint 4475   ` cfv 5888   Fincfn 7955   ficfi 8316

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-fi 8317

This theorem is referenced by:  inelpisys  30217  sigapisys  30218  dynkin  30230