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Theorem iunconALT 38194
 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, iunconALT 38194 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
iunconALT.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
iunconALT.2 ((𝜑𝑘𝐴) → 𝐵𝑋)
iunconALT.3 ((𝜑𝑘𝐴) → 𝑃𝐵)
iunconALT.4 ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Con)
Assertion
Ref Expression
iunconALT (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Con)
Distinct variable groups:   𝜑,𝑘   𝐴,𝑘   𝑘,𝐽   𝑃,𝑘   𝑘,𝑋
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem iunconALT
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 250 . 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 iunconALT.1 . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 iunconALT.2 . 2 ((𝜑𝑘𝐴) → 𝐵𝑋)
4 iunconALT.3 . 2 ((𝜑𝑘𝐴) → 𝑃𝐵)
5 iunconALT.4 . 2 ((𝜑𝑘𝐴) → (𝐽t 𝐵) ∈ Con)
61, 2, 3, 4, 5iunconlem2 38193 1 (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Con)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455  ‘cfv 5804  (class class class)co 6549   ↾t crest 15904  TopOnctopon 20518  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-rest 15906  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-con 21025 This theorem is referenced by: (None)
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