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Theorem conncompconn 22806
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 5022 . . . 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 2761 . . 3 𝑆 = βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦
43oveq2i 7372 . 2 (𝐽 β†Ύt 𝑆) = (𝐽 β†Ύt βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}𝑦)
5 simpl 484 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
6 simpr 486 . . . . . 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 2818 . . . . . . . 8 (π‘₯ = 𝑦 β†’ (𝐴 ∈ π‘₯ ↔ 𝐴 ∈ 𝑦))
8 oveq2 7369 . . . . . . . . 9 (π‘₯ = 𝑦 β†’ (𝐽 β†Ύt π‘₯) = (𝐽 β†Ύt 𝑦))
98eleq1d 2819 . . . . . . . 8 (π‘₯ = 𝑦 β†’ ((𝐽 β†Ύt π‘₯) ∈ Conn ↔ (𝐽 β†Ύt 𝑦) ∈ Conn))
107, 9anbi12d 632 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 β†Ύt 𝑦) ∈ Conn)))
1110elrab 3649 . . . . . 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 496 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ 𝑦 ∈ 𝒫 𝑋)
1413elpwid 4573 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ 𝑦 βŠ† 𝑋)
1512simprd 497 . . . 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 496 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ {π‘₯ ∈ 𝒫 𝑋 ∣ (𝐴 ∈ π‘₯ ∧ (𝐽 β†Ύt π‘₯) ∈ Conn)}) β†’ 𝐴 ∈ 𝑦)
1715simprd 497 . . 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 22802 . 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 2838 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ 𝑋) β†’ (𝐽 β†Ύt 𝑆) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3406  π’« cpw 4564  βˆͺ cuni 4869  βˆͺ ciun 4958  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  TopOnctopon 22282  Conncconn 22785
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-en 8890  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cld 22393  df-conn 22786
This theorem is referenced by:  conncompcld  22808  conncompclo  22809  tgpconncompeqg  23486  tgpconncomp  23487
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