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Theorem conncompid 21282
Description: The connected component containing 𝐴 contains 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompid ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem conncompid
StepHypRef Expression
1 simpr 476 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
21snssd 4372 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ⊆ 𝑋)
3 snex 4938 . . . . . 6 {𝐴} ∈ V
43elpw 4197 . . . . 5 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
52, 4sylibr 224 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ∈ 𝒫 𝑋)
6 snidg 4239 . . . . 5 (𝐴𝑋𝐴 ∈ {𝐴})
76adantl 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ {𝐴})
8 restsn2 21023 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) = 𝒫 {𝐴})
9 pwsn 4460 . . . . . . 7 𝒫 {𝐴} = {∅, {𝐴}}
10 indisconn 21269 . . . . . . 7 {∅, {𝐴}} ∈ Conn
119, 10eqeltri 2726 . . . . . 6 𝒫 {𝐴} ∈ Conn
128, 11syl6eqel 2738 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) ∈ Conn)
137, 12jca 553 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))
14 eleq2 2719 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
15 oveq2 6698 . . . . . . . 8 (𝑥 = {𝐴} → (𝐽t 𝑥) = (𝐽t {𝐴}))
1615eleq1d 2715 . . . . . . 7 (𝑥 = {𝐴} → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t {𝐴}) ∈ Conn))
1714, 16anbi12d 747 . . . . . 6 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn)))
1814, 17anbi12d 747 . . . . 5 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))))
1918rspcev 3340 . . . 4 (({𝐴} ∈ 𝒫 𝑋 ∧ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
205, 7, 13, 19syl12anc 1364 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
21 elunirab 4480 . . 3 (𝐴 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
2220, 21sylibr 224 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
23 conncomp.2 . 2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
2422, 23syl6eleqr 2741 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wrex 2942  {crab 2945  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  {cpr 4212   cuni 4468  cfv 5926  (class class class)co 6690  t crest 16128  TopOnctopon 20763  Conncconn 21262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-rest 16130  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-conn 21263
This theorem is referenced by:  conncompcld  21285  conncompclo  21286  tgpconncompeqg  21962  tgpconncomp  21963
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