Theorem ispisys 30215

Description: The property of being a pi-system. (Contributed by Thierry Arnoux, 10-Jun-2020.)

Hypothesis

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

Assertion

Ref	Expression
ispisys	|- ( S e. P <-> ( S e. ~P ~P O /\ ( fi ` S ) C_ S ) )

Distinct variable groups:   O, s   S, s

Allowed substitution hint:   P( s)

Proof of Theorem ispisys

Step	Hyp	Ref	Expression
1		fveq2 6191	. . 3 |- ( s = S -> ( fi ` s ) = ( fi ` S ) )
2		id 22	. . 3 |- ( s = S -> s = S )
3	1, 2	sseq12d 3634	. 2 |- ( s = S -> ( ( fi ` s ) C_ s <-> ( fi ` S ) C_ S ) )
4		ispisys.p	. 2 |- P = { s e. ~P ~P O | ( fi ` s ) C_ s }
5	3, 4	elrab2 3366	1 |- ( S e. P <-> ( S e. ~P ~P O /\ ( fi ` S ) C_ S ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ~Pcpw 4158   ` cfv 5888   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  ispisys2  30216  sigapildsyslem  30224  sigapildsys  30225  ldgenpisyslem1  30226  ldgenpisyslem3  30228  ldgenpisys  30229