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Theorem conncompconn 23456
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 5063 . . . 4 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦
31, 2eqtri 2763 . . 3 𝑆 = 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦
43oveq2i 7442 . 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 2823 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
8 oveq2 7439 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
98eleq1d 2824 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
107, 9anbi12d 632 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1110elrab 3695 . . . . . 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 4614 . . 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 23452 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦) ∈ Conn)
194, 18eqeltrid 2843 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  𝒫 cpw 4605   cuni 4912   ciun 4996  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
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