Theorem concompid 21044

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

Hypothesis

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

Assertion

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

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

Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem concompid

Step	Hyp	Ref	Expression
1		simpr 476	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
2	1	snssd 4281	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ⊆ 𝑋)
3		snex 4835	. . . . . 6 ⊢ {𝐴} ∈ V
4	3	elpw 4114	. . . . 5 ⊢ ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
5	2, 4	sylibr 223	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → {𝐴} ∈ 𝒫 𝑋)
6		snidg 4153	. . . . 5 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ {𝐴})
7	6	adantl 481	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ {𝐴})
8		restsn2 20785	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) = 𝒫 {𝐴})
9		pwsn 4366	. . . . . . 7 ⊢ 𝒫 {𝐴} = {∅, {𝐴}}
10		indiscon 21031	. . . . . . 7 ⊢ {∅, {𝐴}} ∈ Con
11	9, 10	eqeltri 2684	. . . . . 6 ⊢ 𝒫 {𝐴} ∈ Con
12	8, 11	syl6eqel 2696	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐽 ↾t {𝐴}) ∈ Con)
13	7, 12	jca 553	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Con))
14		eleq2 2677	. . . . . 6 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴}))
15		oveq2 6557	. . . . . . . 8 ⊢ (𝑥 = {𝐴} → (𝐽 ↾t 𝑥) = (𝐽 ↾t {𝐴}))
16	15	eleq1d 2672	. . . . . . 7 ⊢ (𝑥 = {𝐴} → ((𝐽 ↾t 𝑥) ∈ Con ↔ (𝐽 ↾t {𝐴}) ∈ Con))
17	14, 16	anbi12d 743	. . . . . 6 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Con)))
18	14, 17	anbi12d 743	. . . . 5 ⊢ (𝑥 = {𝐴} → ((𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Con))))
19	18	rspcev 3282	. . . 4 ⊢ (({𝐴} ∈ 𝒫 𝑋 ∧ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽 ↾t {𝐴}) ∈ Con))) → ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)))
20	5, 7, 13, 19	syl12anc 1316	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)))
21		elunirab 4384	. . 3 ⊢ (𝐴 ∈ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)} ↔ ∃𝑥 ∈ 𝒫 𝑋(𝐴 ∈ 𝑥 ∧ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)))
22	20, 21	sylibr 223	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)})
23		concomp.2	. 2 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)}
24	22, 23	syl6eleqr 2699	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾_t crest 15904  TopOnctopon 20518  Conccon 21024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-con 21025

This theorem is referenced by:  concompcld  21047  concompclo  21048  tgpconcompeqg  21725  tgpconcomp  21726