Theorem conncompid 22904

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 486	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
2	1	snssd 4808	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋)
3		snex 5427	. . . . . 6 ⊢ {𝐴} ∈ V
4	3	elpw 4602	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
5	2, 4	sylibr 233	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ 𝒫 𝑋)
6		snidg 4658	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴})
7	6	adantl 483	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ {𝐴})
8		restsn2 22644	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) = 𝒫 {𝐴})
9		pwsn 4896	. . . . . . 7 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
10		indisconn 22891	. . . . . . 7 ⊢ {∅, {𝐴}} ∈ Conn
11	9, 10	eqeltri 2830	. . . . . 6 ⊢ 𝒫 {𝐴} ∈ Conn
12	8, 11	eqeltrdi 2842	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) ∈ Conn)
13	7, 12	jca 513	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Conn))
14		eleq2 2823	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
15		oveq2 7404	. . . . . . . 8 ⊢ (𝑥 = {𝐴} → (𝐽 ↾t 𝑥) = (𝐽 ↾t {𝐴}))
16	15	eleq1d 2819	. . . . . . 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 3611	. . . 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 836	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Conn)))
21		elunirab 4920	. . 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 233	. 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 2845	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  {crab 3433   ⊆ wss 3946  ∅c0 4320  𝒫 cpw 4598  {csn 4624  {cpr 4626  ∪ cuni 4904  ‘cfv 6535  (class class class)co 7396   ↾_t crest 17353  TopOnctopon 22381  Conncconn 22884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7843  df-1st 7962  df-2nd 7963  df-en 8928  df-fin 8931  df-fi 9393  df-rest 17355  df-topgen 17376  df-top 22365  df-topon 22382  df-bases 22418  df-cld 22492  df-conn 22885

This theorem is referenced by:  conncompcld  22907  conncompclo  22908  tgpconncompeqg  23585  tgpconncomp  23586