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Theorem iunconnALT 42556
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 42556 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 42556. (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 42555 1 (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  wne 2943  cdif 3884  cun 3885  cin 3886  wss 3887  c0 4256   ciun 4924  cfv 6433  (class class class)co 7275  t crest 17131  TopOnctopon 22059  Conncconn 22562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cld 22170  df-conn 22563
This theorem is referenced by: (None)
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