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Theorem conncompconn 23156
Description: The connected component containing 𝐴 is connected. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
conncomp.2 𝑆 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
Assertion
Ref Expression
conncompconn ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐽   π‘₯,𝑋
Allowed substitution hint:   𝑆(π‘₯)
Proof of Theorem conncompconn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 conncomp.2 . . . 4 𝑆 = βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}
2 uniiun 5060 . . . 4 βˆͺ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} = βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦
31, 2eqtri 2758 . . 3 𝑆 = βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦
43oveq2i 7422 . 2 (𝐽 β†Ύt 𝑆) = (𝐽 β†Ύt βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦)
5 simpl 481 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
6 simpr 483 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)})
7 eleq2w 2815 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (𝐴 ∈ π‘₯ ↔ 𝐴 ∈ 𝑦))
8 oveq2 7419 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (𝐽 β†Ύt π‘₯) = (𝐽 β†Ύt 𝑦))
98eleq1d 2816 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((𝐽 β†Ύt π‘₯) ∈ Conn ↔ (𝐽 β†Ύt 𝑦) ∈ Conn))
107, 9anbi12d 629 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn)))
1110elrab 3682 . . . . . 6 (𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)} ↔ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn)))
126, 11sylib 217 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ (𝑦 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn)))
1312simpld 493 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ 𝑦 ∈ 𝒫 𝑋)
1413elpwid 4610 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ 𝑦 βŠ† 𝑋)
1512simprd 494 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn))
1615simpld 493 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ 𝐴 ∈ 𝑦)
1715simprd 494 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ (𝐽 β†Ύt 𝑦) ∈ Conn)
185, 14, 16, 17iunconn 23152 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦) ∈ Conn)
194, 18eqeltrid 2835 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {crab 3430  π’« cpw 4601  βˆͺ cuni 4907  βˆͺ ciun 4996  β€˜cfv 6542  (class class class)co 7411   β†Ύt crest 17370  TopOnctopon 22632  Conncconn 23135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-cld 22743  df-conn 23136
This theorem is referenced by:  conncompcld  23158  conncompclo  23159  tgpconncompeqg  23836  tgpconncomp  23837
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