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Theorem conncompid 22124
 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 489 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
21snssd 4700 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ⊆ 𝑋)
3 snex 5301 . . . . . 6 {𝐴} ∈ V
43elpw 4499 . . . . 5 ({𝐴} ∈ 𝒫 𝑋 ↔ {𝐴} ⊆ 𝑋)
52, 4sylibr 237 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → {𝐴} ∈ 𝒫 𝑋)
6 snidg 4557 . . . . 5 (𝐴𝑋𝐴 ∈ {𝐴})
76adantl 486 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ {𝐴})
8 restsn2 21864 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) = 𝒫 {𝐴})
9 pwsn 4791 . . . . . . 7 𝒫 {𝐴} = {∅, {𝐴}}
10 indisconn 22111 . . . . . . 7 {∅, {𝐴}} ∈ Conn
119, 10eqeltri 2849 . . . . . 6 𝒫 {𝐴} ∈ Conn
128, 11eqeltrdi 2861 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t {𝐴}) ∈ Conn)
137, 12jca 516 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))
14 eleq2 2841 . . . . . 6 (𝑥 = {𝐴} → (𝐴𝑥𝐴 ∈ {𝐴}))
15 oveq2 7159 . . . . . . . 8 (𝑥 = {𝐴} → (𝐽t 𝑥) = (𝐽t {𝐴}))
1615eleq1d 2837 . . . . . . 7 (𝑥 = {𝐴} → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t {𝐴}) ∈ Conn))
1714, 16anbi12d 634 . . . . . 6 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn)))
1814, 17anbi12d 634 . . . . 5 (𝑥 = {𝐴} → ((𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)) ↔ (𝐴 ∈ {𝐴} ∧ (𝐴 ∈ {𝐴} ∧ (𝐽t {𝐴}) ∈ Conn))))
1918rspcev 3542 . . . 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 4815 . . 3 (𝐴 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ ∃𝑥 ∈ 𝒫 𝑋(𝐴𝑥 ∧ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)))
2220, 21sylibr 237 . 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 2864 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∃wrex 3072  {crab 3075   ⊆ wss 3859  ∅c0 4226  𝒫 cpw 4495  {csn 4523  {cpr 4525  ∪ cuni 4799  ‘cfv 6336  (class class class)co 7151   ↾t crest 16745  TopOnctopon 21603  Conncconn 22104 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-oadd 8117  df-er 8300  df-en 8529  df-fin 8532  df-fi 8901  df-rest 16747  df-topgen 16768  df-top 21587  df-topon 21604  df-bases 21639  df-cld 21712  df-conn 22105 This theorem is referenced by:  conncompcld  22127  conncompclo  22128  tgpconncompeqg  22805  tgpconncomp  22806
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