Theorem unconn 21232

Description: The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)

Assertion

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

Proof of Theorem unconn

Dummy variables x k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. . 3 |- ( ( A i^i B ) =/= (/) <-> E. x x e. ( A i^i B ) )
2		uniiun 4573	. . . . . . . . 9 |- U. { A , B } = U_ k e. { A , B } k
3		simpl1 1064	. . . . . . . . . . . 12 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> J e. (TopOn ` X ) )
4		toponmax 20730	. . . . . . . . . . . 12 |- ( J e. (TopOn ` X ) -> X e. J )
5	3, 4	syl 17	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> X e. J )
6		simpl2l 1114	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> A C_ X )
7	5, 6	ssexd 4805	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> A e. _V )
8		simpl2r 1115	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> B C_ X )
9	5, 8	ssexd 4805	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> B e. _V )
10		uniprg 4450	. . . . . . . . . 10 |- ( ( A e. _V /\ B e. _V ) -> U. { A , B } = ( A u. B ) )
11	7, 9, 10	syl2anc 693	. . . . . . . . 9 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> U. { A , B } = ( A u. B ) )
12	2, 11	syl5eqr 2670	. . . . . . . 8 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> U_ k e. { A , B } k = ( A u. B ) )
13	12	oveq2d 6666	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> ( J ↾t U_ k e. { A , B } k ) = ( J ↾t ( A u. B ) ) )
14		vex 3203	. . . . . . . . . 10 |- k e. _V
15	14	elpr 4198	. . . . . . . . 9 |- ( k e. { A , B } <-> ( k = A \/ k = B ) )
16		simpl2 1065	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> ( A C_ X /\ B C_ X ) )
17		sseq1 3626	. . . . . . . . . . . 12 |- ( k = A -> ( k C_ X <-> A C_ X ) )
18	17	biimprd 238	. . . . . . . . . . 11 |- ( k = A -> ( A C_ X -> k C_ X ) )
19		sseq1 3626	. . . . . . . . . . . 12 |- ( k = B -> ( k C_ X <-> B C_ X ) )
20	19	biimprd 238	. . . . . . . . . . 11 |- ( k = B -> ( B C_ X -> k C_ X ) )
21	18, 20	jaoa 532	. . . . . . . . . 10 |- ( ( k = A \/ k = B ) -> ( ( A C_ X /\ B C_ X ) -> k C_ X ) )
22	16, 21	mpan9 486	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) /\ ( k = A \/ k = B ) ) -> k C_ X )
23	15, 22	sylan2b 492	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) /\ k e. { A , B } ) -> k C_ X )
24		simpl3 1066	. . . . . . . . . . 11 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> x e. ( A i^i B ) )
25		elin 3796	. . . . . . . . . . 11 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
26	24, 25	sylib 208	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> ( x e. A /\ x e. B ) )
27		eleq2 2690	. . . . . . . . . . . 12 |- ( k = A -> ( x e. k <-> x e. A ) )
28	27	biimprd 238	. . . . . . . . . . 11 |- ( k = A -> ( x e. A -> x e. k ) )
29		eleq2 2690	. . . . . . . . . . . 12 |- ( k = B -> ( x e. k <-> x e. B ) )
30	29	biimprd 238	. . . . . . . . . . 11 |- ( k = B -> ( x e. B -> x e. k ) )
31	28, 30	jaoa 532	. . . . . . . . . 10 |- ( ( k = A \/ k = B ) -> ( ( x e. A /\ x e. B ) -> x e. k ) )
32	26, 31	mpan9 486	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) /\ ( k = A \/ k = B ) ) -> x e. k )
33	15, 32	sylan2b 492	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) /\ k e. { A , B } ) -> x e. k )
34		simpr 477	. . . . . . . . . 10 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) )
35		oveq2 6658	. . . . . . . . . . . . 13 |- ( k = A -> ( J ↾t k ) = ( J ↾t A ) )
36	35	eleq1d 2686	. . . . . . . . . . . 12 |- ( k = A -> ( ( J ↾t k ) e. Conn <-> ( J ↾t A ) e. Conn ) )
37	36	biimprd 238	. . . . . . . . . . 11 |- ( k = A -> ( ( J ↾t A ) e. Conn -> ( J ↾t k ) e. Conn ) )
38		oveq2 6658	. . . . . . . . . . . . 13 |- ( k = B -> ( J ↾t k ) = ( J ↾t B ) )
39	38	eleq1d 2686	. . . . . . . . . . . 12 |- ( k = B -> ( ( J ↾t k ) e. Conn <-> ( J ↾t B ) e. Conn ) )
40	39	biimprd 238	. . . . . . . . . . 11 |- ( k = B -> ( ( J ↾t B ) e. Conn -> ( J ↾t k ) e. Conn ) )
41	37, 40	jaoa 532	. . . . . . . . . 10 |- ( ( k = A \/ k = B ) -> ( ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) -> ( J ↾t k ) e. Conn ) )
42	34, 41	mpan9 486	. . . . . . . . 9 |- ( ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) /\ ( k = A \/ k = B ) ) -> ( J ↾t k ) e. Conn )
43	15, 42	sylan2b 492	. . . . . . . 8 |- ( ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) /\ k e. { A , B } ) -> ( J ↾t k ) e. Conn )
44	3, 23, 33, 43	iunconn 21231	. . . . . . 7 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> ( J ↾t U_ k e. { A , B } k ) e. Conn )
45	13, 44	eqeltrrd 2702	. . . . . 6 |- ( ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) /\ ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) ) -> ( J ↾t ( A u. B ) ) e. Conn )
46	45	ex 450	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ x e. ( A i^i B ) ) -> ( ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) -> ( J ↾t ( A u. B ) ) e. Conn ) )
47	46	3expia 1267	. . . 4 |- ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) ) -> ( x e. ( A i^i B ) -> ( ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) -> ( J ↾t ( A u. B ) ) e. Conn ) ) )
48	47	exlimdv 1861	. . 3 |- ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) ) -> ( E. x x e. ( A i^i B ) -> ( ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) -> ( J ↾t ( A u. B ) ) e. Conn ) ) )
49	1, 48	syl5bi 232	. 2 |- ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) ) -> ( ( A i^i B ) =/= (/) -> ( ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) -> ( J ↾t ( A u. B ) ) e. Conn ) ) )
50	49	3impia 1261	1 |- ( ( J e. (TopOn ` X ) /\ ( A C_ X /\ B C_ X ) /\ ( A i^i B ) =/= (/) ) -> ( ( ( J ↾t A ) e. Conn /\ ( J ↾t B ) e. Conn ) -> ( J ↾t ( A u. B ) ) e. Conn ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {cpr 4179   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: (None)