Theorem indispconn 31216

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 20806	. 2 ⊢ {∅, 𝐴} ∈ Top
2		simpl 473	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑥 ∈ ∪ {∅, 𝐴})
3		0ex 4790	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
4		n0i 3920	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ {∅, 𝐴} → ¬ ∪ {∅, 𝐴} = ∅)
5		prprc2 4301	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝐴 ∈ V → {∅, 𝐴} = {∅})
6	5	unieqd 4446	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝐴 ∈ V → ∪ {∅, 𝐴} = ∪ {∅})
7	3	unisn 4451	. . . . . . . . . . . . . . 15 ⊢ ∪ {∅} = ∅
8	6, 7	syl6eq 2672	. . . . . . . . . . . . . 14 ⊢ (¬ 𝐴 ∈ V → ∪ {∅, 𝐴} = ∅)
9	4, 8	nsyl2 142	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∪ {∅, 𝐴} → 𝐴 ∈ V)
10	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝐴 ∈ V)
11		uniprg 4450	. . . . . . . . . . . 12 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
12	3, 10, 11	sylancr 695	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ∪ {∅, 𝐴} = (∅ ∪ 𝐴))
13		uncom 3757	. . . . . . . . . . . 12 ⊢ (∅ ∪ 𝐴) = (𝐴 ∪ ∅)
14		un0 3967	. . . . . . . . . . . 12 ⊢ (𝐴 ∪ ∅) = 𝐴
15	13, 14	eqtri 2644	. . . . . . . . . . 11 ⊢ (∅ ∪ 𝐴) = 𝐴
16	12, 15	syl6eq 2672	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ∪ {∅, 𝐴} = 𝐴)
17	2, 16	eleqtrd 2703	. . . . . . . . 9 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑥 ∈ 𝐴)
18		simpr 477	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑦 ∈ ∪ {∅, 𝐴})
19	18, 16	eleqtrd 2703	. . . . . . . . 9 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → 𝑦 ∈ 𝐴)
20	17, 19	ifcld 4131	. . . . . . . 8 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → if(𝑧 = 0, 𝑥, 𝑦) ∈ 𝐴)
21	20	adantr 481	. . . . . . 7 ⊢ (((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) ∧ 𝑧 ∈ (0[,]1)) → if(𝑧 = 0, 𝑥, 𝑦) ∈ 𝐴)
22		eqid 2622	. . . . . . 7 ⊢ (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))
23	21, 22	fmptd 6385	. . . . . 6 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)):(0[,]1)⟶𝐴)
24		ovex 6678	. . . . . . 7 ⊢ (0[,]1) ∈ V
25		elmapg 7870	. . . . . . 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))
26	10, 24, 25	sylancl 694	. . . . . 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))
27	23, 26	mpbird 247	. . . . 5 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (𝐴 ↑𝑚 (0[,]1)))
28		iitopon 22682	. . . . . 6 ⊢ II ∈ (TopOn‘(0[,]1))
29		cnindis 21096	. . . . . 6 ⊢ ((II ∈ (TopOn‘(0[,]1)) ∧ 𝐴 ∈ V) → (II Cn {∅, 𝐴}) = (𝐴 ↑𝑚 (0[,]1)))
30	28, 10, 29	sylancr 695	. . . . 5 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (II Cn {∅, 𝐴}) = (𝐴 ↑𝑚 (0[,]1)))
31	27, 30	eleqtrrd 2704	. . . 4 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) ∈ (II Cn {∅, 𝐴}))
32		0elunit 12290	. . . . 5 ⊢ 0 ∈ (0[,]1)
33		iftrue 4092	. . . . . 6 ⊢ (𝑧 = 0 → if(𝑧 = 0, 𝑥, 𝑦) = 𝑥)
34		vex 3203	. . . . . 6 ⊢ 𝑥 ∈ V
35	33, 22, 34	fvmpt 6282	. . . . 5 ⊢ (0 ∈ (0[,]1) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥)
36	32, 35	mp1i 13	. . . 4 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0) = 𝑥)
37		1elunit 12291	. . . . 5 ⊢ 1 ∈ (0[,]1)
38		ax-1ne0 10005	. . . . . . . 8 ⊢ 1 ≠ 0
39		neeq1 2856	. . . . . . . 8 ⊢ (𝑧 = 1 → (𝑧 ≠ 0 ↔ 1 ≠ 0))
40	38, 39	mpbiri 248	. . . . . . 7 ⊢ (𝑧 = 1 → 𝑧 ≠ 0)
41		ifnefalse 4098	. . . . . . 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 3203	. . . . . 6 ⊢ 𝑦 ∈ V
44	42, 22, 43	fvmpt 6282	. . . . 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 6190	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (𝑓‘0) = ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘0))
47	46	eqeq1d 2624	. . . . . 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 6190	. . . . . . 7 ⊢ (𝑓 = (𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦)) → (𝑓‘1) = ((𝑧 ∈ (0[,]1) ↦ if(𝑧 = 0, 𝑥, 𝑦))‘1))
49	48	eqeq1d 2624	. . . . . 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 747	. . . . 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 3309	. . . 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	31, 36, 45, 51	syl12anc 1324	. . 3 ⊢ ((𝑥 ∈ ∪ {∅, 𝐴} ∧ 𝑦 ∈ ∪ {∅, 𝐴}) → ∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦))
53	52	rgen2a 2977	. 2 ⊢ ∀𝑥 ∈ ∪ {∅, 𝐴}∀𝑦 ∈ ∪ {∅, 𝐴}∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)
54		eqid 2622	. . 3 ⊢ ∪ {∅, 𝐴} = ∪ {∅, 𝐴}
55	54	ispconn 31205	. 2 ⊢ ({∅, 𝐴} ∈ PConn ↔ ({∅, 𝐴} ∈ Top ∧ ∀𝑥 ∈ ∪ {∅, 𝐴}∀𝑦 ∈ ∪ {∅, 𝐴}∃𝑓 ∈ (II Cn {∅, 𝐴})((𝑓‘0) = 𝑥 ∧ (𝑓‘1) = 𝑦)))
56	1, 53, 55	mpbir2an 955	1 ⊢ {∅, 𝐴} ∈ PConn

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∪ cun 3572  ∅c0 3915  ifcif 4086  {csn 4177  {cpr 4179  ∪ cuni 4436   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↑_𝑚 cmap 7857  0cc0 9936  1c1 9937  [,]cicc 12178  Topctop 20698  TopOnctopon 20715   Cn ccn 21028  IIcii 22678  PConncpconn 31201

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-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-riota 6611  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-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750  df-cn 21031  df-ii 22680  df-pconn 31203

This theorem is referenced by: (None)