Theorem conncompss 21236

Description: The connected component containing A is a superset of any other connected set containing A. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
conncomp.2	|- S = U. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) }

Assertion

Ref	Expression
conncompss	|- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> T C_ S )

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

Allowed substitution hints:   S( x)   T( x)

Proof of Theorem conncompss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. . . . 5 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> T C_ X )
2		conntop 21220	. . . . . . 7 |- ( ( J ↾t T ) e. Conn -> ( J ↾t T ) e. Top )
3	2	3ad2ant3 1084	. . . . . 6 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> ( J ↾t T ) e. Top )
4		restrcl 20961	. . . . . . 7 |- ( ( J ↾t T ) e. Top -> ( J e. _V /\ T e. _V ) )
5	4	simprd 479	. . . . . 6 |- ( ( J ↾t T ) e. Top -> T e. _V )
6		elpwg 4166	. . . . . 6 |- ( T e. _V -> ( T e. ~P X <-> T C_ X ) )
7	3, 5, 6	3syl 18	. . . . 5 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> ( T e. ~P X <-> T C_ X ) )
8	1, 7	mpbird 247	. . . 4 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> T e. ~P X )
9		3simpc 1060	. . . 4 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> ( A e. T /\ ( J ↾t T ) e. Conn ) )
10		eleq2 2690	. . . . . 6 |- ( y = T -> ( A e. y <-> A e. T ) )
11		oveq2 6658	. . . . . . 7 |- ( y = T -> ( J ↾t y ) = ( J ↾t T ) )
12	11	eleq1d 2686	. . . . . 6 |- ( y = T -> ( ( J ↾t y ) e. Conn <-> ( J ↾t T ) e. Conn ) )
13	10, 12	anbi12d 747	. . . . 5 |- ( y = T -> ( ( A e. y /\ ( J ↾t y ) e. Conn ) <-> ( A e. T /\ ( J ↾t T ) e. Conn ) ) )
14		eleq2 2690	. . . . . . 7 |- ( x = y -> ( A e. x <-> A e. y ) )
15		oveq2 6658	. . . . . . . 8 |- ( x = y -> ( J ↾t x ) = ( J ↾t y ) )
16	15	eleq1d 2686	. . . . . . 7 |- ( x = y -> ( ( J ↾t x ) e. Conn <-> ( J ↾t y ) e. Conn ) )
17	14, 16	anbi12d 747	. . . . . 6 |- ( x = y -> ( ( A e. x /\ ( J ↾t x ) e. Conn ) <-> ( A e. y /\ ( J ↾t y ) e. Conn ) ) )
18	17	cbvrabv 3199	. . . . 5 |- { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } = { y e. ~P X | ( A e. y /\ ( J ↾t y ) e. Conn ) }
19	13, 18	elrab2 3366	. . . 4 |- ( T e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } <-> ( T e. ~P X /\ ( A e. T /\ ( J ↾t T ) e. Conn ) ) )
20	8, 9, 19	sylanbrc 698	. . 3 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> T e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } )
21		elssuni 4467	. . 3 |- ( T e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } -> T C_ U. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } )
22	20, 21	syl 17	. 2 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> T C_ U. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } )
23		conncomp.2	. 2 |- S = U. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) }
24	22, 23	syl6sseqr 3652	1 |- ( ( T C_ X /\ A e. T /\ ( J ↾t T ) e. Conn ) -> T C_ S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   U.cuni 4436  (class class class)co 6650   ↾_t crest 16081   Topctop 20698  Conncconn 21214

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-rest 16083  df-top 20699  df-conn 21215

This theorem is referenced by:  conncompcld  21237  tgpconncompeqg  21915  tgpconncomp  21916