Theorem indispconn 32594

Description: The indiscrete topology (or trivial topology) on any set is path-connected. (Contributed by Mario Carneiro, 7-Jul-2015.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
indispconn	⊢ {∅, 𝐴} ∈ PConn

Proof of Theorem indispconn

Dummy variables 𝑥 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		indistop 21607	. 2 ⊢ {∅, 𝐴} ∈ Top
2		simpl 486	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑥 ∈ ∪ {∅, 𝐴})
3		0ex 5175	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
4		n0i 4249	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ {∅, 𝐴} → ¬ ∪ {∅, 𝐴} = ∅)
5		prprc2 4662	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅})
6	5	unieqd 4814	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐴 ∈ V → ∪ {∅, 𝐴} = ∪ {∅})
7	3	unisn 4820	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} = ∅
8	6, 7	eqtrdi 2849	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → ∪ {∅, 𝐴} = ∅)
9	4, 8	nsyl2 143	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∪ {∅, 𝐴} → 𝐴 ∈ V)
10	9	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝐴 ∈ V)
11		uniprg 4818	. . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
12	3, 10, 11	sylancr 590	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
13		uncom 4080	. . . . . . . . . . . 12 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
14		un0 4298	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ ∅) = 𝐴
15	13, 14	eqtri 2821	. . . . . . . . . . 11 ⊢ (∅ ∪ 𝐴) = 𝐴
16	12, 15	eqtrdi 2849	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ∪ {∅, 𝐴} = 𝐴)
17	2, 16	eleqtrd 2892	. . . . . . . . 9 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑥 ∈ 𝐴)
18		simpr 488	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑦 ∈ ∪ {∅, 𝐴})
19	18, 16	eleqtrd 2892	. . . . . . . . 9 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑦 ∈ 𝐴)
20	17, 19	ifcld 4470	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → if(𝑧 = 0, 𝑥, 𝑦) ∈ 𝐴)
21	20	adantr 484	. . . . . . 7 ⊢ (((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) ∧ 𝑧 ∈ (0[,]1)) → if(𝑧 = 0, 𝑥, 𝑦) ∈ 𝐴)
22	21	fmpttd 6856	. . . . . 6 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)):(0[,]1)⟶𝐴)
23		ovex 7168	. . . . . . 7 ⊢ (0[,]1) ∈ V
24		elmapg 8402	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ (0[,]1) ∈ V) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (𝐴 ↑m (0[,]1)) ↔ (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)):(0[,]1)⟶𝐴))
25	10, 23, 24	sylancl 589	. . . . . 6 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (𝐴 ↑m (0[,]1)) ↔ (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)):(0[,]1)⟶𝐴))
26	22, 25	mpbird 260	. . . . 5 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (𝐴 ↑m (0[,]1)))
27		iitopon 23484	. . . . . 6 ⊢ II ∈ (TopOn‘(0[,]1))
28		cnindis 21897	. . . . . 6 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐴 ∈ V) → (II Cn {∅, 𝐴}) = (𝐴 ↑m (0[,]1)))
29	27, 10, 28	sylancr 590	. . . . 5 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (II Cn {∅, 𝐴}) = (𝐴 ↑m (0[,]1)))
30	26, 29	eleqtrrd 2893	. . . 4 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (II Cn {∅, 𝐴}))
31		0elunit 12847	. . . . 5 ⊢ 0 ∈ (0[,]1)
32		iftrue 4431	. . . . . 6 ⊢ (𝑧 = 0 → if(𝑧 = 0, 𝑥, 𝑦) = 𝑥)
33		eqid 2798	. . . . . 6 ⊢ (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))
34		vex 3444	. . . . . 6 ⊢ 𝑥 ∈ V
35	32, 33, 34	fvmpt 6745	. . . . 5 ⊢ (0 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥)
36	31, 35	mp1i 13	. . . 4 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥)
37		1elunit 12848	. . . . 5 ⊢ 1 ∈ (0[,]1)
38		ax-1ne0 10595	. . . . . . . 8 ⊢ 1 ≠ 0
39		neeq1 3049	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧 ≠ 0 ↔ 1 ≠ 0))
40	38, 39	mpbiri 261	. . . . . . 7 ⊢ (𝑧 = 1 → 𝑧 ≠ 0)
41		ifnefalse 4437	. . . . . . 7 ⊢ (𝑧 ≠ 0 → if(𝑧 = 0, 𝑥, 𝑦) = 𝑦)
42	40, 41	syl 17	. . . . . 6 ⊢ (𝑧 = 1 → if(𝑧 = 0, 𝑥, 𝑦) = 𝑦)
43		vex 3444	. . . . . 6 ⊢ 𝑦 ∈ V
44	42, 33, 43	fvmpt 6745	. . . . 5 ⊢ (1 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦)
45	37, 44	mp1i 13	. . . 4 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦)
46		fveq1 6644	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0))
47	46	eqeq1d 2800	. . . . . 6 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → ((𝑓‘0) = 𝑥 ↔ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥))
48		fveq1 6644	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1))
49	48	eqeq1d 2800	. . . . . 6 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → ((𝑓‘1) = 𝑦 ↔ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦))
50	47, 49	anbi12d 633	. . . . 5 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦) ↔ (((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦)))
51	50	rspcev 3571	. . . 4 ⊢ (((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (II Cn {∅, 𝐴}) ∧ (((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥 ∧ ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1) = 𝑦)) → ∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
52	30, 36, 45, 51	syl12anc 835	. . 3 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
53	52	rgen2 3168	. 2 ⊢ ∀𝑥 ∈ ∪ {∅, 𝐴}∀𝑦 ∈ ∪ {∅, 𝐴}∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)
54		eqid 2798	. . 3 ⊢ ∪ {∅, 𝐴} = ∪ {∅, 𝐴}
55	54	ispconn 32583	. 2 ⊢ ({∅, 𝐴} ∈ PConn ↔ ({∅, 𝐴} ∈ Top ∧ ∀𝑥 ∈ ∪ {∅, 𝐴}∀𝑦 ∈ ∪ {∅, 𝐴}∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
56	1, 53, 55	mpbir2an 710	1 ⊢ {∅, 𝐴} ∈ PConn

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∪ cun 3879  ∅c0 4243  ifcif 4425  {csn 4525  {cpr 4527  ∪ cuni 4800   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  0cc0 10526  1c1 10527  [,]cicc 12729  Topctop 21498  TopOnctopon 21515   Cn ccn 21829  IIcii 23480  PConncpconn 32579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-sup 8890  df-inf 8891  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-icc 12733  df-seq 13365  df-exp 13426  df-cj 14450  df-re 14451  df-im 14452  df-sqrt 14586  df-abs 14587  df-topgen 16709  df-psmet 20083  df-xmet 20084  df-met 20085  df-bl 20086  df-mopn 20087  df-top 21499  df-topon 21516  df-bases 21551  df-cn 21832  df-ii 23482  df-pconn 32581

This theorem is referenced by: (None)