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Theorem unconn 23453
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 4359 . . 3 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 uniiun 5063 . . . . . . . . 9 {𝐴, 𝐵} = 𝑘 ∈ {𝐴, 𝐵}𝑘
3 simpl1 1190 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐽 ∈ (TopOn‘𝑋))
4 toponmax 22948 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
53, 4syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑋𝐽)
6 simpl2l 1225 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐴𝑋)
75, 6ssexd 5330 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐴 ∈ V)
8 simpl2r 1226 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐵𝑋)
95, 8ssexd 5330 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10 uniprg 4928 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
117, 9, 10syl2anc 584 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → {𝐴, 𝐵} = (𝐴𝐵))
122, 11eqtr3id 2789 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑘 ∈ {𝐴, 𝐵}𝑘 = (𝐴𝐵))
1312oveq2d 7447 . . . . . . 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 4655 . . . . . . . . 9 (𝑘 ∈ {𝐴, 𝐵} ↔ (𝑘 = 𝐴𝑘 = 𝐵))
16 simpl2 1191 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐴𝑋𝐵𝑋))
17 sseq1 4021 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑘𝑋𝐴𝑋))
1817biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝐴𝑋𝑘𝑋))
19 sseq1 4021 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑘𝑋𝐵𝑋))
2019biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝐵𝑋𝑘𝑋))
2118, 20jaoa 957 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝐴𝑋𝐵𝑋) → 𝑘𝑋))
2216, 21mpan9 506 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑘𝑋)
2315, 22sylan2b 594 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑘𝑋)
24 simpl3 1192 . . . . . . . . . . 11 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑥 ∈ (𝐴𝐵))
25 elin 3979 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2624, 25sylib 218 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝑥𝐴𝑥𝐵))
27 eleq2 2828 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑥𝑘𝑥𝐴))
2827biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝑥𝐴𝑥𝑘))
29 eleq2 2828 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑥𝑘𝑥𝐵))
3029biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝑥𝐵𝑥𝑘))
3128, 30jaoa 957 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝑥𝐴𝑥𝐵) → 𝑥𝑘))
3226, 31mpan9 506 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑥𝑘)
3315, 32sylan2b 594 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑥𝑘)
34 simpr 484 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn))
35 oveq2 7439 . . . . . . . . . . . . 13 (𝑘 = 𝐴 → (𝐽t 𝑘) = (𝐽t 𝐴))
3635eleq1d 2824 . . . . . . . . . . . 12 (𝑘 = 𝐴 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐴) ∈ Conn))
3736biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐴 → ((𝐽t 𝐴) ∈ Conn → (𝐽t 𝑘) ∈ Conn))
38 oveq2 7439 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐽t 𝑘) = (𝐽t 𝐵))
3938eleq1d 2824 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐵) ∈ Conn))
4039biimprd 248 . . . . . . . . . . 11 (𝑘 = 𝐵 → ((𝐽t 𝐵) ∈ Conn → (𝐽t 𝑘) ∈ Conn))
4137, 40jaoa 957 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t 𝑘) ∈ Conn))
4234, 41mpan9 506 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → (𝐽t 𝑘) ∈ Conn)
4315, 42sylan2b 594 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → (𝐽t 𝑘) ∈ Conn)
443, 23, 33, 43iunconn 23452 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t 𝑘 ∈ {𝐴, 𝐵}𝑘) ∈ Conn)
4513, 44eqeltrrd 2840 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t (𝐴𝐵)) ∈ Conn)
4645ex 412 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn))
47463expia 1120 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝑥 ∈ (𝐴𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
4847exlimdv 1931 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (∃𝑥 𝑥 ∈ (𝐴𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
491, 48biimtrid 242 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐴𝐵) ≠ ∅ → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
50493impia 1116 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 395  wo 847  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  Vcvv 3478  cun 3961  cin 3962  wss 3963  c0 4339  {cpr 4633   cuni 4912   ciun 4996  cfv 6563  (class class class)co 7431  t crest 17467  TopOnctopon 22932  Conncconn 23435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-en 8985  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-conn 23436
This theorem is referenced by: (None)
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