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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 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
Assertion
Ref Expression
concompid ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem concompid
StepHypRef Expression
1 simpr 476 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
21snssd 4281 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ⊆ 𝑋)
3 snex 4835 . . . . . 6 {𝐴} ∈ V
43elpw 4114 . . . . 5 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
52, 4sylibr 223 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ∈ 𝒫 𝑋)
6 snidg 4153 . . . . 5 (𝐴𝑋𝐴 ∈ {𝐴})
76adantl 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ {𝐴})
8 restsn2 20785 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) = 𝒫 {𝐴})
9 pwsn 4366 . . . . . . 7 𝒫 {𝐴} = {∅, {𝐴}}
10 indiscon 21031 . . . . . . 7 {∅, {𝐴}} ∈ Con
119, 10eqeltri 2684 . . . . . 6 𝒫 {𝐴} ∈ Con
128, 11syl6eqel 2696 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) ∈ Con)
137, 12jca 553 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Con))
14 eleq2 2677 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
15 oveq2 6557 . . . . . . . 8 (𝑥 = {𝐴} → (𝐽t 𝑥) = (𝐽t {𝐴}))
1615eleq1d 2672 . . . . . . 7 (𝑥 = {𝐴} → ((𝐽t 𝑥) ∈ Con ↔ (𝐽t {𝐴}) ∈ Con))
1714, 16anbi12d 743 . . . . . 6 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Con)))
1814, 17anbi12d 743 . . . . 5 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Con))))
1918rspcev 3282 . . . 4 (({𝐴} ∈ 𝒫 𝑋 ∧ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Con))) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)))
205, 7, 13, 19syl12anc 1316 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)))
21 elunirab 4384 . . 3 (𝐴 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)} ↔ ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)))
2220, 21sylibr 223 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
23 concomp.2 . 2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
2422, 23syl6eleqr 2699 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
 Colors of variables: wff setvar class 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
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