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Theorem iunconnALT 39932
 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 http://us.metamath.org/other/completeusersproof/iunconaltvd.html. As it is verified by the Metamath program, iunconnALT 39932 verifies http://us.metamath.org/other/completeusersproof/iunconaltvd.html. (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 253 . 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 39931 1 (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   ∈ wcel 2157   ≠ wne 2971   ∖ cdif 3766   ∪ cun 3767   ∩ cin 3768   ⊆ wss 3769  ∅c0 4115  ∪ ciun 4710  ‘cfv 6101  (class class class)co 6878   ↾t crest 16396  TopOnctopon 21043  Conncconn 21543 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-oadd 7803  df-er 7982  df-en 8196  df-fin 8199  df-fi 8559  df-rest 16398  df-topgen 16419  df-top 21027  df-topon 21044  df-bases 21079  df-cld 21152  df-conn 21544 This theorem is referenced by: (None)
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