MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  connsuba Structured version   Visualization version   GIF version

Theorem connsuba 21204
Description: Connectedness for a subspace. See connsub 21205. (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 20946 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
2 dfconn2 21203 . . 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 3198 . . . . 5 𝑥 ∈ V
54inex1 4790 . . . 4 (𝑥𝐴) ∈ V
65a1i 11 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
7 toponmax 20711 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
87adantr 481 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋𝐽)
9 simpr 477 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
108, 9ssexd 4796 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
11 elrest 16069 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
1210, 11syldan 487 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
13 vex 3198 . . . . . 6 𝑦 ∈ V
1413inex1 4790 . . . . 5 (𝑦𝐴) ∈ V
1514a1i 11 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑦𝐽) → (𝑦𝐴) ∈ V)
16 elrest 16069 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
1710, 16syldan 487 . . . . 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 791 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑢 = (𝑥𝐴))
2019neeq1d 2850 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢 ≠ ∅ ↔ (𝑥𝐴) ≠ ∅))
21 simpr 477 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑣 = (𝑦𝐴))
2221neeq1d 2850 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦𝐴) ≠ ∅))
2319, 21ineq12d 3807 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∩ (𝑦𝐴)))
24 inindir 3823 . . . . . . . 8 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∩ (𝑦𝐴))
2523, 24syl6eqr 2672 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
2625eqeq1d 2622 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) = ∅ ↔ ((𝑥𝑦) ∩ 𝐴) = ∅))
2720, 22, 263anbi123d 1397 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅)))
2819, 21uneq12d 3760 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∪ (𝑦𝐴)))
29 indir 3867 . . . . . . 7 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∪ (𝑦𝐴))
3028, 29syl6eqr 2672 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
3130neeq1d 2850 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) ≠ 𝐴 ↔ ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴))
3227, 31imbi12d 334 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴) ↔ (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
3315, 18, 32ralxfr2d 4873 . . 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 4873 . 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 268 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 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  Vcvv 3195  cun 3565  cin 3566  wss 3567  c0 3907  cfv 5876  (class class class)co 6635  t crest 16062  TopOnctopon 20696  Conncconn 21195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-oadd 7549  df-er 7727  df-en 7941  df-fin 7944  df-fi 8302  df-rest 16064  df-topgen 16085  df-top 20680  df-topon 20697  df-bases 20731  df-cld 20804  df-conn 21196
This theorem is referenced by:  connsub  21205  nconnsubb  21207
  Copyright terms: Public domain W3C validator