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Theorem iunconnALT 44956
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 44956 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 44956. (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 261 . 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 44955 1 (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wne 2940  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333   ciun 4991  cfv 6561  (class class class)co 7431  t crest 17465  TopOnctopon 22916  Conncconn 23419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-en 8986  df-fin 8989  df-fi 9451  df-rest 17467  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-conn 23420
This theorem is referenced by: (None)
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