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Theorem connsuba 21956
Description: Connectedness for a subspace. See connsub 21957. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
Assertion
Ref Expression
connsuba ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((𝐽t 𝐴) ∈ Conn ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐽,𝑦   𝑥,𝑋,𝑦

Proof of Theorem connsuba
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resttopon 21697 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
2 dfconn2 21955 . . 3 ((𝐽t 𝐴) ∈ (TopOn‘𝐴) → ((𝐽t 𝐴) ∈ Conn ↔ ∀𝑢 ∈ (𝐽t 𝐴)∀𝑣 ∈ (𝐽t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴)))
31, 2syl 17 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((𝐽t 𝐴) ∈ Conn ↔ ∀𝑢 ∈ (𝐽t 𝐴)∀𝑣 ∈ (𝐽t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴)))
4 vex 3495 . . . . 5 𝑥 ∈ V
54inex1 5212 . . . 4 (𝑥𝐴) ∈ V
65a1i 11 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
7 toponmax 21462 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
87adantr 481 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋𝐽)
9 simpr 485 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
108, 9ssexd 5219 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
11 elrest 16689 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
1210, 11syldan 591 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
13 vex 3495 . . . . . 6 𝑦 ∈ V
1413inex1 5212 . . . . 5 (𝑦𝐴) ∈ V
1514a1i 11 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑦𝐽) → (𝑦𝐴) ∈ V)
16 elrest 16689 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
1710, 16syldan 591 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
1817adantr 481 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
19 simplr 765 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑢 = (𝑥𝐴))
2019neeq1d 3072 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢 ≠ ∅ ↔ (𝑥𝐴) ≠ ∅))
21 simpr 485 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑣 = (𝑦𝐴))
2221neeq1d 3072 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦𝐴) ≠ ∅))
2319, 21ineq12d 4187 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∩ (𝑦𝐴)))
24 inindir 4201 . . . . . . . 8 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∩ (𝑦𝐴))
2523, 24syl6eqr 2871 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
2625eqeq1d 2820 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) = ∅ ↔ ((𝑥𝑦) ∩ 𝐴) = ∅))
2720, 22, 263anbi123d 1427 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅)))
2819, 21uneq12d 4137 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∪ (𝑦𝐴)))
29 indir 4249 . . . . . . 7 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∪ (𝑦𝐴))
3028, 29syl6eqr 2871 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
3130neeq1d 3072 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) ≠ 𝐴 ↔ ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴))
3227, 31imbi12d 346 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴) ↔ (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
3315, 18, 32ralxfr2d 5301 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) → (∀𝑣 ∈ (𝐽t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴) ↔ ∀𝑦𝐽 (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
346, 12, 33ralxfr2d 5301 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (∀𝑢 ∈ (𝐽t 𝐴)∀𝑣 ∈ (𝐽t 𝐴)((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴) ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
353, 34bitrd 280 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → ((𝐽t 𝐴) ∈ Conn ↔ ∀𝑥𝐽𝑦𝐽 (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  Vcvv 3492  cun 3931  cin 3932  wss 3933  c0 4288  cfv 6348  (class class class)co 7145  t crest 16682  TopOnctopon 21446  Conncconn 21947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-oadd 8095  df-er 8278  df-en 8498  df-fin 8501  df-fi 8863  df-rest 16684  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-conn 21948
This theorem is referenced by:  connsub  21957  nconnsubb  21959
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