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Theorem connsuba 22723
Description: Connectedness for a subspace. See connsub 22724. (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 22464 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
2 dfconn2 22722 . . 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 3448 . . . . 5 𝑥 ∈ V
54inex1 5273 . . . 4 (𝑥𝐴) ∈ V
65a1i 11 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
7 toponmax 22227 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
87adantr 482 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋𝐽)
9 simpr 486 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
108, 9ssexd 5280 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
11 elrest 17269 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
1210, 11syldan 592 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
13 vex 3448 . . . . . 6 𝑦 ∈ V
1413inex1 5273 . . . . 5 (𝑦𝐴) ∈ V
1514a1i 11 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑦𝐽) → (𝑦𝐴) ∈ V)
16 elrest 17269 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
1710, 16syldan 592 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
1817adantr 482 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
19 simplr 768 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑢 = (𝑥𝐴))
2019neeq1d 3002 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢 ≠ ∅ ↔ (𝑥𝐴) ≠ ∅))
21 simpr 486 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑣 = (𝑦𝐴))
2221neeq1d 3002 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦𝐴) ≠ ∅))
2319, 21ineq12d 4172 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∩ (𝑦𝐴)))
24 inindir 4186 . . . . . . . 8 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∩ (𝑦𝐴))
2523, 24eqtr4di 2796 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
2625eqeq1d 2740 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) = ∅ ↔ ((𝑥𝑦) ∩ 𝐴) = ∅))
2720, 22, 263anbi123d 1437 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅)))
2819, 21uneq12d 4123 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∪ (𝑦𝐴)))
29 indir 4234 . . . . . . 7 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∪ (𝑦𝐴))
3028, 29eqtr4di 2796 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
3130neeq1d 3002 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) ≠ 𝐴 ↔ ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴))
3227, 31imbi12d 345 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴) ↔ (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
3315, 18, 32ralxfr2d 5364 . . 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 5364 . 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 279 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2942  wral 3063  wrex 3072  Vcvv 3444  cun 3907  cin 3908  wss 3909  c0 4281  cfv 6494  (class class class)co 7352  t crest 17262  TopOnctopon 22211  Conncconn 22714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-int 4907  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5530  df-eprel 5536  df-po 5544  df-so 5545  df-fr 5587  df-we 5589  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7796  df-1st 7914  df-2nd 7915  df-en 8843  df-fin 8846  df-fi 9306  df-rest 17264  df-topgen 17285  df-top 22195  df-topon 22212  df-bases 22248  df-cld 22322  df-conn 22715
This theorem is referenced by:  connsub  22724  nconnsubb  22726
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