Theorem connsub 21224

Description: Two equivalent ways of saying that a subspace topology is connected. (Contributed by Jeff Hankins, 9-Jul-2009.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Assertion

Ref	Expression
connsub	|- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> ( ( J ↾t S ) e. Conn <-> A. x e. J A. y e. J ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( x i^i y ) C_ ( X \ S ) ) -> -. S C_ ( x u. y ) ) ) )

Distinct variable groups:   x, y, J   x, S, y   x, X, y

Proof of Theorem connsub

Step	Hyp	Ref	Expression
1		connsuba 21223	. 2 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> ( ( J ↾t S ) e. Conn <-> A. x e. J A. y e. J ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( ( x i^i y ) i^i S ) = (/) ) -> ( ( x u. y ) i^i S ) =/= S ) ) )
2		inss1 3833	. . . . . . 7 |- ( x i^i y ) C_ x
3		toponss 20731	. . . . . . . 8 |- ( ( J e. (TopOn ` X ) /\ x e. J ) -> x C_ X )
4	3	ad2ant2r 783	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> x C_ X )
5	2, 4	syl5ss 3614	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> ( x i^i y ) C_ X )
6		reldisj 4020	. . . . . 6 |- ( ( x i^i y ) C_ X -> ( ( ( x i^i y ) i^i S ) = (/) <-> ( x i^i y ) C_ ( X \ S ) ) )
7	5, 6	syl 17	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> ( ( ( x i^i y ) i^i S ) = (/) <-> ( x i^i y ) C_ ( X \ S ) ) )
8	7	3anbi3d 1405	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( ( x i^i y ) i^i S ) = (/) ) <-> ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( x i^i y ) C_ ( X \ S ) ) ) )
9		sseqin2 3817	. . . . . . 7 |- ( S C_ ( x u. y ) <-> ( ( x u. y ) i^i S ) = S )
10	9	a1i 11	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> ( S C_ ( x u. y ) <-> ( ( x u. y ) i^i S ) = S ) )
11	10	bicomd 213	. . . . 5 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> ( ( ( x u. y ) i^i S ) = S <-> S C_ ( x u. y ) ) )
12	11	necon3abid 2830	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> ( ( ( x u. y ) i^i S ) =/= S <-> -. S C_ ( x u. y ) ) )
13	8, 12	imbi12d 334	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ S C_ X ) /\ ( x e. J /\ y e. J ) ) -> ( ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( ( x i^i y ) i^i S ) = (/) ) -> ( ( x u. y ) i^i S ) =/= S ) <-> ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( x i^i y ) C_ ( X \ S ) ) -> -. S C_ ( x u. y ) ) ) )
14	13	2ralbidva 2988	. 2 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> ( A. x e. J A. y e. J ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( ( x i^i y ) i^i S ) = (/) ) -> ( ( x u. y ) i^i S ) =/= S ) <-> A. x e. J A. y e. J ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( x i^i y ) C_ ( X \ S ) ) -> -. S C_ ( x u. y ) ) ) )
15	1, 14	bitrd 268	1 |- ( ( J e. (TopOn ` X ) /\ S C_ X ) -> ( ( J ↾t S ) e. Conn <-> A. x e. J A. y e. J ( ( ( x i^i S ) =/= (/) /\ ( y i^i S ) =/= (/) /\ ( x i^i y ) C_ ( X \ S ) ) -> -. S C_ ( x u. y ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   ` 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:  iunconn  21231  clsconn  21233  reconn  22631  iunconnlem2  39171