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Theorem connsubclo 22008
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
connsubclo.1 𝑋 = 𝐽
connsubclo.3 (𝜑𝐴𝑋)
connsubclo.4 (𝜑 → (𝐽t 𝐴) ∈ Conn)
connsubclo.5 (𝜑𝐵𝐽)
connsubclo.6 (𝜑 → (𝐵𝐴) ≠ ∅)
connsubclo.7 (𝜑𝐵 ∈ (Clsd‘𝐽))
Assertion
Ref Expression
connsubclo (𝜑𝐴𝐵)

Proof of Theorem connsubclo
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2821 . . . 4 (𝐽t 𝐴) = (𝐽t 𝐴)
2 connsubclo.4 . . . 4 (𝜑 → (𝐽t 𝐴) ∈ Conn)
3 connsubclo.7 . . . . . 6 (𝜑𝐵 ∈ (Clsd‘𝐽))
4 cldrcl 21610 . . . . . 6 (𝐵 ∈ (Clsd‘𝐽) → 𝐽 ∈ Top)
53, 4syl 17 . . . . 5 (𝜑𝐽 ∈ Top)
6 connsubclo.1 . . . . . . . 8 𝑋 = 𝐽
76topopn 21490 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
85, 7syl 17 . . . . . 6 (𝜑𝑋𝐽)
9 connsubclo.3 . . . . . 6 (𝜑𝐴𝑋)
108, 9ssexd 5201 . . . . 5 (𝜑𝐴 ∈ V)
11 connsubclo.5 . . . . 5 (𝜑𝐵𝐽)
12 elrestr 16681 . . . . 5 ((𝐽 ∈ Top ∧ 𝐴 ∈ V ∧ 𝐵𝐽) → (𝐵𝐴) ∈ (𝐽t 𝐴))
135, 10, 11, 12syl3anc 1368 . . . 4 (𝜑 → (𝐵𝐴) ∈ (𝐽t 𝐴))
14 connsubclo.6 . . . 4 (𝜑 → (𝐵𝐴) ≠ ∅)
15 eqid 2821 . . . . . 6 (𝐵𝐴) = (𝐵𝐴)
16 ineq1 4156 . . . . . . 7 (𝑥 = 𝐵 → (𝑥𝐴) = (𝐵𝐴))
1716rspceeqv 3615 . . . . . 6 ((𝐵 ∈ (Clsd‘𝐽) ∧ (𝐵𝐴) = (𝐵𝐴)) → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴))
183, 15, 17sylancl 589 . . . . 5 (𝜑 → ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴))
196restcld 21756 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐴𝑋) → ((𝐵𝐴) ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴)))
205, 9, 19syl2anc 587 . . . . 5 (𝜑 → ((𝐵𝐴) ∈ (Clsd‘(𝐽t 𝐴)) ↔ ∃𝑥 ∈ (Clsd‘𝐽)(𝐵𝐴) = (𝑥𝐴)))
2118, 20mpbird 260 . . . 4 (𝜑 → (𝐵𝐴) ∈ (Clsd‘(𝐽t 𝐴)))
221, 2, 13, 14, 21connclo 21999 . . 3 (𝜑 → (𝐵𝐴) = (𝐽t 𝐴))
236restuni 21746 . . . 4 ((𝐽 ∈ Top ∧ 𝐴𝑋) → 𝐴 = (𝐽t 𝐴))
245, 9, 23syl2anc 587 . . 3 (𝜑𝐴 = (𝐽t 𝐴))
2522, 24eqtr4d 2859 . 2 (𝜑 → (𝐵𝐴) = 𝐴)
26 sseqin2 4167 . 2 (𝐴𝐵 ↔ (𝐵𝐴) = 𝐴)
2725, 26sylibr 237 1 (𝜑𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  wne 3007  wrex 3127  Vcvv 3471  cin 3909  wss 3910  c0 4266   cuni 4811  cfv 6328  (class class class)co 7130  t crest 16673  Topctop 21477  Clsdccld 21600  Conncconn 21995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6121  df-ord 6167  df-on 6168  df-lim 6169  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-om 7556  df-1st 7664  df-2nd 7665  df-wrecs 7922  df-recs 7983  df-rdg 8021  df-oadd 8081  df-er 8264  df-en 8485  df-fin 8488  df-fi 8851  df-rest 16675  df-topgen 16696  df-top 21478  df-topon 21495  df-bases 21530  df-cld 21603  df-conn 21996
This theorem is referenced by:  conncn  22010  conncompclo  22019
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