Theorem conncompid 22039

Description: The connected component containing 𝐴 contains 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)

Hypothesis

Ref	Expression
conncomp.2	⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}

Assertion

Ref	Expression
conncompid	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)

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

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem conncompid

Step	Hyp	Ref	Expression
1		simpr 487	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
2	1	snssd 4742	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋)
3		snex 5332	. . . . . 6 ⊢ {𝐴} ∈ V
4	3	elpw 4543	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
5	2, 4	sylibr 236	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ 𝒫 𝑋)
6		snidg 4599	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴})
7	6	adantl 484	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ {𝐴})
8		restsn2 21779	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) = 𝒫 {𝐴})
9		pwsn 4830	. . . . . . 7 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
10		indisconn 22026	. . . . . . 7 ⊢ {∅, {𝐴}} ∈ Conn
11	9, 10	eqeltri 2909	. . . . . 6 ⊢ 𝒫 {𝐴} ∈ Conn
12	8, 11	eqeltrdi 2921	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) ∈ Conn)
13	7, 12	jca 514	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))
14		eleq2 2901	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
15		oveq2 7164	. . . . . . . 8 ⊢ (𝑥 = {𝐴} → (𝐽 ↾t 𝑥) = (𝐽 ↾t {𝐴}))
16	15	eleq1d 2897	. . . . . . 7 ⊢ (𝑥 = {𝐴} → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t {𝐴}) ∈ Conn))
17	14, 16	anbi12d 632	. . . . . 6 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn)))
18	14, 17	anbi12d 632	. . . . 5 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))))
19	18	rspcev 3623	. . . 4 ⊢ (({𝐴} ∈ 𝒫 𝑋 ∧ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))) → ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)))
20	5, 7, 13, 19	syl12anc 834	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)))
21		elunirab 4854	. . 3 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)} ↔ ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)))
22	20, 21	sylibr 236	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)})
23		conncomp.2	. 2 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)}
24	22, 23	eleqtrrdi 2924	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  {crab 3142   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4567  {cpr 4569  ∪ cuni 4838  ‘cfv 6355  (class class class)co 7156   ↾_t crest 16694  TopOnctopon 21518  Conncconn 22019

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-oadd 8106  df-er 8289  df-en 8510  df-fin 8513  df-fi 8875  df-rest 16696  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cld 21627  df-conn 22020

This theorem is referenced by:  conncompcld  22042  conncompclo  22043  tgpconncompeqg  22720  tgpconncomp  22721