Theorem conncompid 23779

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 490	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
2	1	snssd 4747	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋)
3		snex 5401	. . . . . 6 ⊢ {𝐴} ∈ V
4	3	elpw 4564	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
5	2, 4	sylibr 237	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ 𝒫 𝑋)
6		snidg 4621	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴})
7	6	adantl 487	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ {𝐴})
8		restsn2 23519	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) = 𝒫 {𝐴})
9		pwsn 4830	. . . . . . 7 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
10		indisconn 23766	. . . . . . 7 ⊢ {∅, {𝐴}} ∈ Conn
11	9, 10	eqeltri 2860	. . . . . 6 ⊢ 𝒫 {𝐴} ∈ Conn
12	8, 11	eqeltrdi 2872	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) ∈ Conn)
13	7, 12	jca 521	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))
14		eleq2 2853	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
15		oveq2 7451	. . . . . . . 8 ⊢ (𝑥 = {𝐴} → (𝐽 ↾t 𝑥) = (𝐽 ↾t {𝐴}))
16	15	eleq1d 2849	. . . . . . 7 ⊢ (𝑥 = {𝐴} → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t {𝐴}) ∈ Conn))
17	14, 16	anbi12d 644	. . . . . 6 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn)))
18	14, 17	anbi12d 644	. . . . 5 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))))
19	18	rspcev 3580	. . . 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 850	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)))
21		elunirab 4885	. . 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 237	. 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 2875	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090  {crab 3416   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588  {cpr 4590  ∪ cuni 4870  ‘cfv 6558  (class class class)co 7443   ↾_t crest 17620  TopOnctopon 23258  Conncconn 23759

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-en 9002  df-fin 9005  df-fi 9430  df-rest 17622  df-topgen 17643  df-top 23242  df-topon 23259  df-bases 23294  df-cld 23367  df-conn 23760

This theorem is used by:  conncompcld  23782  conncompclo  23783  tgpconncompeqg  24461  tgpconncomp  24462