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

Theorem unconn 23377
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 4346 . . 3 ((𝐴𝐵) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴𝐵))
2 uniiun 5062 . . . . . . . . 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 22872 . . . . . . . . . . . 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 5325 . . . . . . . . . 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 5325 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝐵 ∈ V)
10 uniprg 4925 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝐴, 𝐵} = (𝐴𝐵))
117, 9, 10syl2anc 582 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → {𝐴, 𝐵} = (𝐴𝐵))
122, 11eqtr3id 2779 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → 𝑘 ∈ {𝐴, 𝐵}𝑘 = (𝐴𝐵))
1312oveq2d 7435 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t 𝑘 ∈ {𝐴, 𝐵}𝑘) = (𝐽t (𝐴𝐵)))
14 vex 3465 . . . . . . . . . 10 𝑘 ∈ V
1514elpr 4654 . . . . . . . . 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 4002 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑘𝑋𝐴𝑋))
1817biimprd 247 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝐴𝑋𝑘𝑋))
19 sseq1 4002 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑘𝑋𝐵𝑋))
2019biimprd 247 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝐵𝑋𝑘𝑋))
2118, 20jaoa 953 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝐴𝑋𝐵𝑋) → 𝑘𝑋))
2216, 21mpan9 505 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑘𝑋)
2315, 22sylan2b 592 . . . . . . . 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 3960 . . . . . . . . . . 11 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
2624, 25sylib 217 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝑥𝐴𝑥𝐵))
27 eleq2 2814 . . . . . . . . . . . 12 (𝑘 = 𝐴 → (𝑥𝑘𝑥𝐴))
2827biimprd 247 . . . . . . . . . . 11 (𝑘 = 𝐴 → (𝑥𝐴𝑥𝑘))
29 eleq2 2814 . . . . . . . . . . . 12 (𝑘 = 𝐵 → (𝑥𝑘𝑥𝐵))
3029biimprd 247 . . . . . . . . . . 11 (𝑘 = 𝐵 → (𝑥𝐵𝑥𝑘))
3128, 30jaoa 953 . . . . . . . . . 10 ((𝑘 = 𝐴𝑘 = 𝐵) → ((𝑥𝐴𝑥𝐵) → 𝑥𝑘))
3226, 31mpan9 505 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → 𝑥𝑘)
3315, 32sylan2b 592 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝑥𝑘)
34 simpr 483 . . . . . . . . . 10 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn))
35 oveq2 7427 . . . . . . . . . . . . 13 (𝑘 = 𝐴 → (𝐽t 𝑘) = (𝐽t 𝐴))
3635eleq1d 2810 . . . . . . . . . . . 12 (𝑘 = 𝐴 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐴) ∈ Conn))
3736biimprd 247 . . . . . . . . . . 11 (𝑘 = 𝐴 → ((𝐽t 𝐴) ∈ Conn → (𝐽t 𝑘) ∈ Conn))
38 oveq2 7427 . . . . . . . . . . . . 13 (𝑘 = 𝐵 → (𝐽t 𝑘) = (𝐽t 𝐵))
3938eleq1d 2810 . . . . . . . . . . . 12 (𝑘 = 𝐵 → ((𝐽t 𝑘) ∈ Conn ↔ (𝐽t 𝐵) ∈ Conn))
4039biimprd 247 . . . . . . . . . . 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 505 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ (𝑘 = 𝐴𝑘 = 𝐵)) → (𝐽t 𝑘) ∈ Conn)
4315, 42sylan2b 592 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) ∧ 𝑘 ∈ {𝐴, 𝐵}) → (𝐽t 𝑘) ∈ Conn)
443, 23, 33, 43iunconn 23376 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t 𝑘 ∈ {𝐴, 𝐵}𝑘) ∈ Conn)
4513, 44eqeltrrd 2826 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋) ∧ 𝑥 ∈ (𝐴𝐵)) ∧ ((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn)) → (𝐽t (𝐴𝐵)) ∈ Conn)
4645ex 411 . . . . 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 1928 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (∃𝑥 𝑥 ∈ (𝐴𝐵) → (((𝐽t 𝐴) ∈ Conn ∧ (𝐽t 𝐵) ∈ Conn) → (𝐽t (𝐴𝐵)) ∈ Conn)))
491, 48biimtrid 241 . 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 394  wo 845  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2929  Vcvv 3461  cun 3942  cin 3943  wss 3944  c0 4322  {cpr 4632   cuni 4909   ciun 4997  cfv 6549  (class class class)co 7419  t crest 17405  TopOnctopon 22856  Conncconn 23359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-en 8965  df-fin 8968  df-fi 9436  df-rest 17407  df-topgen 17428  df-top 22840  df-topon 22857  df-bases 22893  df-cld 22967  df-conn 23360
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator