Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iunconnALT Structured version   Visualization version   GIF version

Theorem iunconnALT 44649
Description: The indexed union of connected overlapping subspaces sharing a common point is connected. This proof was automatically derived by completeusersproof from its Virtual Deduction proof counterpart https://us.metamath.org/other/completeusersproof/iunconaltvd.html. As it is verified by the Metamath program, iunconnALT 44649 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 44649. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
iunconnALT.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
iunconnALT.2 ((𝜑𝑘𝐴) → 𝐵𝑋)
iunconnALT.3 ((𝜑𝑘𝐴) → 𝑃𝐵)
iunconnALT.4 ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)
Assertion
Ref Expression
iunconnALT (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
Distinct variable groups:   𝜑,𝑘   𝐴,𝑘   𝑘,𝐽   𝑃,𝑘   𝑘,𝑋
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem iunconnALT
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 260 . 2 (((((((𝜑𝑢𝐽) ∧ 𝑣𝐽) ∧ (𝑢 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑣 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑢𝑣) ⊆ (𝑋 𝑘𝐴 𝐵)) ∧ 𝑘𝐴 𝐵 ⊆ (𝑢𝑣)) ↔ ((((((𝜑𝑢𝐽) ∧ 𝑣𝐽) ∧ (𝑢 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑣 𝑘𝐴 𝐵) ≠ ∅) ∧ (𝑢𝑣) ⊆ (𝑋 𝑘𝐴 𝐵)) ∧ 𝑘𝐴 𝐵 ⊆ (𝑢𝑣)))
2 iunconnALT.1 . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 iunconnALT.2 . 2 ((𝜑𝑘𝐴) → 𝐵𝑋)
4 iunconnALT.3 . 2 ((𝜑𝑘𝐴) → 𝑃𝐵)
5 iunconnALT.4 . 2 ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Conn)
61, 2, 3, 4, 5iunconnlem2 44648 1 (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2099  wne 2930  cdif 3943  cun 3944  cin 3945  wss 3946  c0 4322   ciun 4993  cfv 6546  (class class class)co 7416  t crest 17430  TopOnctopon 22900  Conncconn 23403
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7995  df-2nd 7996  df-en 8967  df-fin 8970  df-fi 9447  df-rest 17432  df-topgen 17453  df-top 22884  df-topon 22901  df-bases 22937  df-cld 23011  df-conn 23404
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator