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Theorem conncompconn 23441
Description: The connected component containing 𝐴 is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
Assertion
Ref Expression
conncompconn ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hint:   𝑆(𝑥)

Proof of Theorem conncompconn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 conncomp.2 . . . 4 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
2 uniiun 5057 . . . 4 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦
31, 2eqtri 2764 . . 3 𝑆 = 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦
43oveq2i 7443 . 2 (𝐽t 𝑆) = (𝐽t 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦)
5 simpl 482 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6 simpr 484 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
7 eleq2w 2824 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
8 oveq2 7440 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
98eleq1d 2825 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
107, 9anbi12d 632 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1110elrab 3691 . . . . . 6 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
126, 11sylib 218 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1312simpld 494 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝑦 ∈ 𝒫 𝑋)
1413elpwid 4608 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝑦𝑋)
1512simprd 495 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))
1615simpld 494 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝐴𝑦)
1715simprd 495 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → (𝐽t 𝑦) ∈ Conn)
185, 14, 16, 17iunconn 23437 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦) ∈ Conn)
194, 18eqeltrid 2844 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  {crab 3435  𝒫 cpw 4599   cuni 4906   ciun 4990  cfv 6560  (class class class)co 7432  t crest 17466  TopOnctopon 22917  Conncconn 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-en 8987  df-fin 8990  df-fi 9452  df-rest 17468  df-topgen 17489  df-top 22901  df-topon 22918  df-bases 22954  df-cld 23028  df-conn 23421
This theorem is referenced by:  conncompcld  23443  conncompclo  23444  tgpconncompeqg  24121  tgpconncomp  24122
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