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Theorem concompss 21046
 Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.)
Hypothesis
Ref Expression
concomp.2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
Assertion
Ref Expression
concompss ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → 𝑇𝑆)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐽   𝑥,𝑋
Allowed substitution hints:   𝑆(𝑥)   𝑇(𝑥)

Proof of Theorem concompss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simp1 1054 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → 𝑇𝑋)
2 contop 21030 . . . . . . 7 ((𝐽t 𝑇) ∈ Con → (𝐽t 𝑇) ∈ Top)
323ad2ant3 1077 . . . . . 6 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → (𝐽t 𝑇) ∈ Top)
4 restrcl 20771 . . . . . . 7 ((𝐽t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V))
54simprd 478 . . . . . 6 ((𝐽t 𝑇) ∈ Top → 𝑇 ∈ V)
6 elpwg 4116 . . . . . 6 (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
73, 5, 63syl 18 . . . . 5 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → (𝑇 ∈ 𝒫 𝑋𝑇𝑋))
81, 7mpbird 246 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → 𝑇 ∈ 𝒫 𝑋)
9 3simpc 1053 . . . 4 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con))
10 eleq2 2677 . . . . . 6 (𝑦 = 𝑇 → (𝐴𝑦𝐴𝑇))
11 oveq2 6557 . . . . . . 7 (𝑦 = 𝑇 → (𝐽t 𝑦) = (𝐽t 𝑇))
1211eleq1d 2672 . . . . . 6 (𝑦 = 𝑇 → ((𝐽t 𝑦) ∈ Con ↔ (𝐽t 𝑇) ∈ Con))
1310, 12anbi12d 743 . . . . 5 (𝑦 = 𝑇 → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Con) ↔ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con)))
14 eleq2 2677 . . . . . . 7 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
15 oveq2 6557 . . . . . . . 8 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
1615eleq1d 2672 . . . . . . 7 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Con ↔ (𝐽t 𝑦) ∈ Con))
1714, 16anbi12d 743 . . . . . 6 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Con)))
1817cbvrabv 3172 . . . . 5 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Con)}
1913, 18elrab2 3333 . . . 4 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con)))
208, 9, 19sylanbrc 695 . . 3 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
21 elssuni 4403 . . 3 (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)} → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
2220, 21syl 17 . 2 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → 𝑇 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)})
23 concomp.2 . 2 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Con)}
2422, 23syl6sseqr 3615 1 ((𝑇𝑋𝐴𝑇 ∧ (𝐽t 𝑇) ∈ Con) → 𝑇𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-rest 15906  df-top 20521  df-con 21025 This theorem is referenced by:  concompcld  21047  tgpconcompeqg  21725  tgpconcomp  21726
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