Theorem conncompid 23549

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 489	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
2	1	snssd 4748	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋)
3		snex 5401	. . . . . 6 ⊢ {𝐴} ∈ V
4	3	elpw 4562	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
5	2, 4	sylibr 237	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ 𝒫 𝑋)
6		snidg 4622	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴})
7	6	adantl 486	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ {𝐴})
8		restsn2 23289	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) = 𝒫 {𝐴})
9		pwsn 4861	. . . . . . 7 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
10		indisconn 23536	. . . . . . 7 ⊢ {∅, {𝐴}} ∈ Conn
11	9, 10	eqeltri 2861	. . . . . 6 ⊢ 𝒫 {𝐴} ∈ Conn
12	8, 11	eqeltrdi 2873	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) ∈ Conn)
13	7, 12	jca 520	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))
14		eleq2 2854	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
15		oveq2 7408	. . . . . . . 8 ⊢ (𝑥 = {𝐴} → (𝐽 ↾t 𝑥) = (𝐽 ↾t {𝐴}))
16	15	eleq1d 2850	. . . . . . 7 ⊢ (𝑥 = {𝐴} → ((𝐽 ↾t 𝑥) ∈ Conn ↔ (𝐽 ↾t {𝐴}) ∈ Conn))
17	14, 16	anbi12d 643	. . . . . 6 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn)))
18	14, 17	anbi12d 643	. . . . 5 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))))
19	18	rspcev 3584	. . . 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 849	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)))
21		elunirab 4883	. . 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 2876	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∃wrex 3089  {crab 3417   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558  {csn 4585  {cpr 4587  ∪ cuni 4868  ‘cfv 6525  (class class class)co 7400   ↾_t crest 17463  TopOnctopon 23028  Conncconn 23529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-en 8932  df-fin 8935  df-fi 9359  df-rest 17465  df-topgen 17486  df-top 23012  df-topon 23029  df-bases 23064  df-cld 23137  df-conn 23530

This theorem is referenced by:  conncompcld  23552  conncompclo  23553  tgpconncompeqg  24230  tgpconncomp  24231