Theorem connsuba 21223

Description: Connectedness for a subspace. See connsub 21224. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)

Assertion

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

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

Proof of Theorem connsuba

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		resttopon 20965	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( J ↾t A ) e. (TopOn ` A ) )
2		dfconn2 21222	. . 3 |- ( ( J ↾t A ) e. (TopOn ` A ) -> ( ( J ↾t A ) e. Conn <-> A. u e. ( J ↾t A ) A. v e. ( J ↾t A ) ( ( u =/= (/) /\ v =/= (/) /\ ( u i^i v ) = (/) ) -> ( u u. v ) =/= A ) ) )
3	1, 2	syl 17	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( ( J ↾t A ) e. Conn <-> A. u e. ( J ↾t A ) A. v e. ( J ↾t A ) ( ( u =/= (/) /\ v =/= (/) /\ ( u i^i v ) = (/) ) -> ( u u. v ) =/= A ) ) )
4		vex 3203	. . . . 5 |- x e. _V
5	4	inex1 4799	. . . 4 |- ( x i^i A ) e. _V
6	5	a1i 11	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ x e. J ) -> ( x i^i A ) e. _V )
7		toponmax 20730	. . . . . 6 |- ( J e. (TopOn ` X ) -> X e. J )
8	7	adantr 481	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> X e. J )
9		simpr 477	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A C_ X )
10	8, 9	ssexd 4805	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> A e. _V )
11		elrest 16088	. . . 4 |- ( ( J e. (TopOn ` X ) /\ A e. _V ) -> ( u e. ( J ↾t A ) <-> E. x e. J u = ( x i^i A ) ) )
12	10, 11	syldan 487	. . 3 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( u e. ( J ↾t A ) <-> E. x e. J u = ( x i^i A ) ) )
13		vex 3203	. . . . . 6 |- y e. _V
14	13	inex1 4799	. . . . 5 |- ( y i^i A ) e. _V
15	14	a1i 11	. . . 4 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ y e. J ) -> ( y i^i A ) e. _V )
16		elrest 16088	. . . . . 6 |- ( ( J e. (TopOn ` X ) /\ A e. _V ) -> ( v e. ( J ↾t A ) <-> E. y e. J v = ( y i^i A ) ) )
17	10, 16	syldan 487	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( v e. ( J ↾t A ) <-> E. y e. J v = ( y i^i A ) ) )
18	17	adantr 481	. . . 4 |- ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) -> ( v e. ( J ↾t A ) <-> E. y e. J v = ( y i^i A ) ) )
19		simplr 792	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> u = ( x i^i A ) )
20	19	neeq1d 2853	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( u =/= (/) <-> ( x i^i A ) =/= (/) ) )
21		simpr 477	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> v = ( y i^i A ) )
22	21	neeq1d 2853	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( v =/= (/) <-> ( y i^i A ) =/= (/) ) )
23	19, 21	ineq12d 3815	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( u i^i v ) = ( ( x i^i A ) i^i ( y i^i A ) ) )
24		inindir 3831	. . . . . . . 8 |- ( ( x i^i y ) i^i A ) = ( ( x i^i A ) i^i ( y i^i A ) )
25	23, 24	syl6eqr 2674	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( u i^i v ) = ( ( x i^i y ) i^i A ) )
26	25	eqeq1d 2624	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( ( u i^i v ) = (/) <-> ( ( x i^i y ) i^i A ) = (/) ) )
27	20, 22, 26	3anbi123d 1399	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( ( u =/= (/) /\ v =/= (/) /\ ( u i^i v ) = (/) ) <-> ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) ) )
28	19, 21	uneq12d 3768	. . . . . . 7 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( u u. v ) = ( ( x i^i A ) u. ( y i^i A ) ) )
29		indir 3875	. . . . . . 7 |- ( ( x u. y ) i^i A ) = ( ( x i^i A ) u. ( y i^i A ) )
30	28, 29	syl6eqr 2674	. . . . . 6 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( u u. v ) = ( ( x u. y ) i^i A ) )
31	30	neeq1d 2853	. . . . 5 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( ( u u. v ) =/= A <-> ( ( x u. y ) i^i A ) =/= A ) )
32	27, 31	imbi12d 334	. . . 4 |- ( ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) /\ v = ( y i^i A ) ) -> ( ( ( u =/= (/) /\ v =/= (/) /\ ( u i^i v ) = (/) ) -> ( u u. v ) =/= A ) <-> ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) ) )
33	15, 18, 32	ralxfr2d 4882	. . 3 |- ( ( ( J e. (TopOn ` X ) /\ A C_ X ) /\ u = ( x i^i A ) ) -> ( A. v e. ( J ↾t A ) ( ( u =/= (/) /\ v =/= (/) /\ ( u i^i v ) = (/) ) -> ( u u. v ) =/= A ) <-> A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) ) )
34	6, 12, 33	ralxfr2d 4882	. 2 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( A. u e. ( J ↾t A ) A. v e. ( J ↾t A ) ( ( u =/= (/) /\ v =/= (/) /\ ( u i^i v ) = (/) ) -> ( u u. v ) =/= A ) <-> A. x e. J A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) ) )
35	3, 34	bitrd 268	1 |- ( ( J e. (TopOn ` X ) /\ A C_ X ) -> ( ( J ↾t A ) e. Conn <-> A. x e. J A. y e. J ( ( ( x i^i A ) =/= (/) /\ ( y i^i A ) =/= (/) /\ ( ( x i^i y ) i^i A ) = (/) ) -> ( ( x u. y ) i^i A ) =/= A ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   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:  connsub  21224  nconnsubb  21226