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Theorem connsuba 21346
Description: Connectedness for a subspace. See connsub 21347. (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 21088 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝐽t 𝐴) ∈ (TopOn‘𝐴))
2 dfconn2 21345 . . 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 3307 . . . . 5 𝑥 ∈ V
54inex1 4907 . . . 4 (𝑥𝐴) ∈ V
65a1i 11 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑥𝐽) → (𝑥𝐴) ∈ V)
7 toponmax 20853 . . . . . 6 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
87adantr 472 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝑋𝐽)
9 simpr 479 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴𝑋)
108, 9ssexd 4913 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → 𝐴 ∈ V)
11 elrest 16211 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
1210, 11syldan 488 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑢 ∈ (𝐽t 𝐴) ↔ ∃𝑥𝐽 𝑢 = (𝑥𝐴)))
13 vex 3307 . . . . . 6 𝑦 ∈ V
1413inex1 4907 . . . . 5 (𝑦𝐴) ∈ V
1514a1i 11 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑦𝐽) → (𝑦𝐴) ∈ V)
16 elrest 16211 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ V) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
1710, 16syldan 488 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
1817adantr 472 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) → (𝑣 ∈ (𝐽t 𝐴) ↔ ∃𝑦𝐽 𝑣 = (𝑦𝐴)))
19 simplr 809 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑢 = (𝑥𝐴))
2019neeq1d 2955 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢 ≠ ∅ ↔ (𝑥𝐴) ≠ ∅))
21 simpr 479 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → 𝑣 = (𝑦𝐴))
2221neeq1d 2955 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑣 ≠ ∅ ↔ (𝑦𝐴) ≠ ∅))
2319, 21ineq12d 3923 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∩ (𝑦𝐴)))
24 inindir 3939 . . . . . . . 8 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∩ (𝑦𝐴))
2523, 24syl6eqr 2776 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
2625eqeq1d 2726 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) = ∅ ↔ ((𝑥𝑦) ∩ 𝐴) = ∅))
2720, 22, 263anbi123d 1512 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) ↔ ((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅)))
2819, 21uneq12d 3876 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝐴) ∪ (𝑦𝐴)))
29 indir 3983 . . . . . . 7 ((𝑥𝑦) ∩ 𝐴) = ((𝑥𝐴) ∪ (𝑦𝐴))
3028, 29syl6eqr 2776 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (𝑢𝑣) = ((𝑥𝑦) ∩ 𝐴))
3130neeq1d 2955 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → ((𝑢𝑣) ≠ 𝐴 ↔ ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴))
3227, 31imbi12d 333 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴𝑋) ∧ 𝑢 = (𝑥𝐴)) ∧ 𝑣 = (𝑦𝐴)) → (((𝑢 ≠ ∅ ∧ 𝑣 ≠ ∅ ∧ (𝑢𝑣) = ∅) → (𝑢𝑣) ≠ 𝐴) ↔ (((𝑥𝐴) ≠ ∅ ∧ (𝑦𝐴) ≠ ∅ ∧ ((𝑥𝑦) ∩ 𝐴) = ∅) → ((𝑥𝑦) ∩ 𝐴) ≠ 𝐴)))
3315, 18, 32ralxfr2d 4987 . . 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 4987 . 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 383  w3a 1072   = wceq 1596  wcel 2103  wne 2896  wral 3014  wrex 3015  Vcvv 3304  cun 3678  cin 3679  wss 3680  c0 4023  cfv 6001  (class class class)co 6765  t crest 16204  TopOnctopon 20838  Conncconn 21337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-oadd 7684  df-er 7862  df-en 8073  df-fin 8076  df-fi 8433  df-rest 16206  df-topgen 16227  df-top 20822  df-topon 20839  df-bases 20873  df-cld 20946  df-conn 21338
This theorem is referenced by:  connsub  21347  nconnsubb  21349
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