MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  conncompid Structured version   Visualization version   GIF version

Theorem conncompid 23455
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 484 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
21snssd 4814 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ⊆ 𝑋)
3 snex 5442 . . . . . 6 {𝐴} ∈ V
43elpw 4609 . . . . 5 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
52, 4sylibr 234 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ∈ 𝒫 𝑋)
6 snidg 4665 . . . . 5 (𝐴𝑋𝐴 ∈ {𝐴})
76adantl 481 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ {𝐴})
8 restsn2 23195 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) = 𝒫 {𝐴})
9 pwsn 4905 . . . . . . 7 𝒫 {𝐴} = {∅, {𝐴}}
10 indisconn 23442 . . . . . . 7 {∅, {𝐴}} ∈ Conn
119, 10eqeltri 2835 . . . . . 6 𝒫 {𝐴} ∈ Conn
128, 11eqeltrdi 2847 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) ∈ Conn)
137, 12jca 511 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))
14 eleq2 2828 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
15 oveq2 7439 . . . . . . . 8 (𝑥 = {𝐴} → (𝐽t 𝑥) = (𝐽t {𝐴}))
1615eleq1d 2824 . . . . . . 7 (𝑥 = {𝐴} → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t {𝐴}) ∈ Conn))
1714, 16anbi12d 632 . . . . . 6 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn)))
1814, 17anbi12d 632 . . . . 5 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))))
1918rspcev 3622 . . . 4 (({𝐴} ∈ 𝒫 𝑋 ∧ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
205, 7, 13, 19syl12anc 837 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
21 elunirab 4927 . . 3 (𝐴 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
2220, 21sylibr 234 . 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, 23eleqtrrdi 2850 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  {cpr 4633   cuni 4912  cfv 6563  (class class class)co 7431  t crest 17467  TopOnctopon 22932  Conncconn 23435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-conn 23436
This theorem is referenced by:  conncompcld  23458  conncompclo  23459  tgpconncompeqg  24136  tgpconncomp  24137
  Copyright terms: Public domain W3C validator