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Theorem consubclo 21037
 Description: If a clopen set meets a connected subspace, it must contain the entire subspace. (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
consubclo.1 𝑋 = 𝐽
consubclo.3 (𝜑𝐴𝑋)
consubclo.4 (𝜑 → (𝐽t 𝐴) ∈ Con)
consubclo.5 (𝜑𝐵𝐽)
consubclo.6 (𝜑 → (𝐵𝐴) ≠ ∅)
consubclo.7 (𝜑𝐵 ∈ (Clsd‘𝐽))
Assertion
Ref Expression
consubclo (𝜑𝐴𝐵)

Proof of Theorem consubclo
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . 4 (𝐽t 𝐴) = (𝐽t 𝐴)
2 consubclo.4 . . . 4 (𝜑 → (𝐽t 𝐴) ∈ Con)
3 consubclo.7 . . . . . 6 (𝜑𝐵 ∈ (Clsd‘𝐽))
4 cldrcl 20640 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
53, 4syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
6 consubclo.1 . . . . . . . 8 𝑋 = 𝐽
76topopn 20536 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
85, 7syl 17 . . . . . 6 (𝜑𝑋𝐽)
9 consubclo.3 . . . . . 6 (𝜑𝐴𝑋)
108, 9ssexd 4733 . . . . 5 (𝜑𝐴 ∈ V)
11 consubclo.5 . . . . 5 (𝜑𝐵𝐽)
12 elrestr 15912 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵𝐽) → (𝐵𝐴) ∈ (𝐽t 𝐴))
135, 10, 11, 12syl3anc 1318 . . . 4 (𝜑 → (𝐵𝐴) ∈ (𝐽t 𝐴))
14 consubclo.6 . . . 4 (𝜑 → (𝐵𝐴) ≠ ∅)
15 eqid 2610 . . . . . 6 (𝐵𝐴) = (𝐵𝐴)
16 ineq1 3769 . . . . . . . 8 (𝑥 = 𝐵 → (𝑥𝐴) = (𝐵𝐴))
1716eqeq2d 2620 . . . . . . 7 (𝑥 = 𝐵 → ((𝐵𝐴) = (𝑥𝐴) ↔ (𝐵𝐴) = (𝐵𝐴)))
1817rspcev 3282 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵𝐴) = (𝐵𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴))
193, 15, 18sylancl 693 . . . . 5 (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴))
206restcld 20786 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐵𝐴) ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴)))
215, 9, 20syl2anc 691 . . . . 5 (𝜑 → ((𝐵𝐴) ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴)))
2219, 21mpbird 246 . . . 4 (𝜑 → (𝐵𝐴) ∈ (Clsd‘(𝐽t 𝐴)))
231, 2, 13, 14, 22conclo 21028 . . 3 (𝜑 → (𝐵𝐴) = (𝐽t 𝐴))
246restuni 20776 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
255, 9, 24syl2anc 691 . . 3 (𝜑𝐴 = (𝐽t 𝐴))
2623, 25eqtr4d 2647 . 2 (𝜑 → (𝐵𝐴) = 𝐴)
27 sseqin2 3779 . 2 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
2826, 27sylibr 223 1 (𝜑𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ cuni 4372  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  Topctop 20517  Clsdccld 20630  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-3or 1032  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-con 21025 This theorem is referenced by:  concn  21039  concompclo  21048
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