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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 ∣ (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 486 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
21snssd 4808 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ⊆ 𝑋)
3 snex 5427 . . . . . 6 {𝐴} ∈ V
43elpw 4602 . . . . 5 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
52, 4sylibr 233 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ∈ 𝒫 𝑋)
6 snidg 4658 . . . . 5 (𝐴𝑋𝐴 ∈ {𝐴})
76adantl 483 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ {𝐴})
8 restsn2 22644 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) = 𝒫 {𝐴})
9 pwsn 4896 . . . . . . 7 𝒫 {𝐴} = {∅, {𝐴}}
10 indisconn 22891 . . . . . . 7 {∅, {𝐴}} ∈ Conn
119, 10eqeltri 2830 . . . . . 6 𝒫 {𝐴} ∈ Conn
128, 11eqeltrdi 2842 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) ∈ Conn)
137, 12jca 513 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))
14 eleq2 2823 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
15 oveq2 7404 . . . . . . . 8 (𝑥 = {𝐴} → (𝐽t 𝑥) = (𝐽t {𝐴}))
1615eleq1d 2819 . . . . . . 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 3611 . . . 4 (({𝐴} ∈ 𝒫 𝑋 ∧ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
205, 7, 13, 19syl12anc 836 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
21 elunirab 4920 . . 3 (𝐴 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
2220, 21sylibr 233 . 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 2845 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
Colors of variables: wff setvar class
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
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