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Theorem unconn 22023
Description: The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)
Assertion
Ref Expression
unconn ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ≠ ∅) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))

Proof of Theorem unconn
Dummy variables 𝑥 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4291 . . 3 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 uniiun 4963 . . . . . . . . 9 {𝐴, 𝐵} = 𝑘 ∈ {𝐴, 𝐵}𝑘
3 simpl1 1188 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 21520 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
53, 4syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑋𝐽)
6 simpl2l 1223 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐴𝑋)
75, 6ssexd 5209 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐴 ∈ V)
8 simpl2r 1224 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐵𝑋)
95, 8ssexd 5209 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10 uniprg 4837 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
117, 9, 10syl2anc 587 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → {𝐴, 𝐵} = (𝐴𝐵))
122, 11syl5eqr 2873 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑘 ∈ {𝐴, 𝐵}𝑘 = (𝐴𝐵))
1312oveq2d 7154 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t 𝑘 ∈ {𝐴, 𝐵}𝑘) = (𝐽t (𝐴𝐵)))
14 vex 3482 . . . . . . . . . 10 𝑘 ∈ V
1514elpr 4571 . . . . . . . . 9 (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴𝑘 = 𝐵))
16 simpl2 1189 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐴𝑋𝐵𝑋))
17 sseq1 3976 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑘𝑋𝐴𝑋))
1817biimprd 251 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝐴𝑋𝑘𝑋))
19 sseq1 3976 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑘𝑋𝐵𝑋))
2019biimprd 251 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝐵𝑋𝑘𝑋))
2118, 20jaoa 953 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝐴𝑋𝐵𝑋) → 𝑘𝑋))
2216, 21mpan9 510 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑘𝑋)
2315, 22sylan2b 596 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑘𝑋)
24 simpl3 1190 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑥 ∈ (𝐴𝐵))
25 elin 3934 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2624, 25sylib 221 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝑥𝐴𝑥𝐵))
27 eleq2 2904 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑥𝑘𝑥𝐴))
2827biimprd 251 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝑥𝐴𝑥𝑘))
29 eleq2 2904 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑥𝑘𝑥𝐵))
3029biimprd 251 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝑥𝐵𝑥𝑘))
3128, 30jaoa 953 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝑥𝐴𝑥𝐵) → 𝑥𝑘))
3226, 31mpan9 510 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑥𝑘)
3315, 32sylan2b 596 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑥𝑘)
34 simpr 488 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn))
35 oveq2 7146 . . . . . . . . . . . . 13 (𝑘 = 𝐴 → (𝐽t 𝑘) = (𝐽t 𝐴))
3635eleq1d 2900 . . . . . . . . . . . 12 (𝑘 = 𝐴 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐴) ∈ Conn))
3736biimprd 251 . . . . . . . . . . 11 (𝑘 = 𝐴 → ((𝐽t 𝐴) ∈ Conn → (𝐽t 𝑘) ∈ Conn))
38 oveq2 7146 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐽t 𝑘) = (𝐽t 𝐵))
3938eleq1d 2900 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐵) ∈ Conn))
4039biimprd 251 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐽t 𝐵) ∈ Conn → (𝐽t 𝑘) ∈ Conn))
4137, 40jaoa 953 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t 𝑘) ∈ Conn))
4234, 41mpan9 510 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → (𝐽t 𝑘) ∈ Conn)
4315, 42sylan2b 596 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → (𝐽t 𝑘) ∈ Conn)
443, 23, 33, 43iunconn 22022 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t 𝑘 ∈ {𝐴, 𝐵}𝑘) ∈ Conn)
4513, 44eqeltrrd 2917 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t (𝐴𝐵)) ∈ Conn)
4645ex 416 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))
47463expia 1118 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝑥 ∈ (𝐴𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
4847exlimdv 1935 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (∃𝑥 𝑥 ∈ (𝐴𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
491, 48syl5bi 245 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐵) ≠ ∅ → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
50493impia 1114 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐴𝐵) ≠ ∅) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 844  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wne 3013  Vcvv 3479  cun 3916  cin 3917  wss 3918  c0 4274  {cpr 4550   cuni 4819   ciun 4900  cfv 6336  (class class class)co 7138  t crest 16683  TopOnctopon 21504  Conncconn 22005
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 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444
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 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-oadd 8089  df-er 8272  df-en 8493  df-fin 8496  df-fi 8859  df-rest 16685  df-topgen 16706  df-top 21488  df-topon 21505  df-bases 21540  df-cld 21613  df-conn 22006
This theorem is referenced by: (None)
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