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Theorem conncompconn 21435
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 4723 . . . 4 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦
31, 2eqtri 2780 . . 3 𝑆 = 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦
43oveq2i 6822 . 2 (𝐽t 𝑆) = (𝐽t 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦)
5 simpl 474 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
6 simpr 479 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
7 eleq2w 2821 . . . . . . . 8 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
8 oveq2 6819 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
98eleq1d 2822 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
107, 9anbi12d 749 . . . . . . 7 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1110elrab 3502 . . . . . 6 (𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
126, 11sylib 208 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
1312simpld 477 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝑦 ∈ 𝒫 𝑋)
1413elpwid 4312 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝑦𝑋)
1512simprd 482 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))
1615simpld 477 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → 𝐴𝑦)
1715simprd 482 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}) → (𝐽t 𝑦) ∈ Conn)
185, 14, 16, 17iunconn 21431 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦) ∈ Conn)
194, 18syl5eqel 2841 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1630  wcel 2137  {crab 3052  𝒫 cpw 4300   cuni 4586   ciun 4670  cfv 6047  (class class class)co 6811  t crest 16281  TopOnctopon 20915  Conncconn 21414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-oadd 7731  df-er 7909  df-en 8120  df-fin 8123  df-fi 8480  df-rest 16283  df-topgen 16304  df-top 20899  df-topon 20916  df-bases 20950  df-cld 21023  df-conn 21415
This theorem is referenced by:  conncompcld  21437  conncompclo  21438  tgpconncompeqg  22114  tgpconncomp  22115
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