Theorem conncompconn 21235

Description: The connected component containing A is connected. (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
conncompconn	|- ( ( J e. (TopOn ` X ) /\ A e. X ) -> ( J ↾t S ) e. Conn )

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

Allowed substitution hint:   S( x)

Proof of Theorem conncompconn

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		conncomp.2	. . . 4 |- S = U. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) }
2		uniiun 4573	. . . 4 |- U. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } = U_ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } y
3	1, 2	eqtri 2644	. . 3 |- S = U_ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } y
4	3	oveq2i 6661	. 2 |- ( J ↾t S ) = ( J ↾t U_ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } y )
5		simpl 473	. . 3 |- ( ( J e. (TopOn ` X ) /\ A e. X ) -> J e. (TopOn ` X ) )
6		simpr 477	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ A e. X ) /\ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } ) -> y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } )
7		eleq2 2690	. . . . . . . 8 |- ( x = y -> ( A e. x <-> A e. y ) )
8		oveq2 6658	. . . . . . . . 9 |- ( x = y -> ( J ↾t x ) = ( J ↾t y ) )
9	8	eleq1d 2686	. . . . . . . 8 |- ( x = y -> ( ( J ↾t x ) e. Conn <-> ( J ↾t y ) e. Conn ) )
10	7, 9	anbi12d 747	. . . . . . 7 |- ( x = y -> ( ( A e. x /\ ( J ↾t x ) e. Conn ) <-> ( A e. y /\ ( J ↾t y ) e. Conn ) ) )
11	10	elrab 3363	. . . . . 6 |- ( y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } <-> ( y e. ~P X /\ ( A e. y /\ ( J ↾t y ) e. Conn ) ) )
12	6, 11	sylib 208	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ A e. X ) /\ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } ) -> ( y e. ~P X /\ ( A e. y /\ ( J ↾t y ) e. Conn ) ) )
13	12	simpld 475	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ A e. X ) /\ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } ) -> y e. ~P X )
14	13	elpwid 4170	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ A e. X ) /\ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } ) -> y C_ X )
15	12	simprd 479	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ A e. X ) /\ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } ) -> ( A e. y /\ ( J ↾t y ) e. Conn ) )
16	15	simpld 475	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ A e. X ) /\ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } ) -> A e. y )
17	15	simprd 479	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ A e. X ) /\ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } ) -> ( J ↾t y ) e. Conn )
18	5, 14, 16, 17	iunconn 21231	. 2 |- ( ( J e. (TopOn ` X ) /\ A e. X ) -> ( J ↾t U_ y e. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Conn ) } y ) e. Conn )
19	4, 18	syl5eqel 2705	1 |- ( ( J e. (TopOn ` X ) /\ A e. X ) -> ( J ↾t S ) e. Conn )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   ~Pcpw 4158   U.cuni 4436   U_ciun 4520   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081  TopOnctopon 20715  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-3or 1038  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-rest 16083  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-conn 21215

This theorem is referenced by:  conncompcld  21237  conncompclo  21238  tgpconncompeqg  21915  tgpconncomp  21916