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Theorem iunconnALT 42094
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 42094 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 42094. (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 264 . 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 42093 1 (𝜑 → (𝐽t 𝑘𝐴 𝐵) ∈ Conn)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2114  wne 2934  cdif 3840  cun 3841  cin 3842  wss 3843  c0 4211   ciun 4881  cfv 6339  (class class class)co 7170  t crest 16797  TopOnctopon 21661  Conncconn 22162
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-en 8556  df-fin 8559  df-fi 8948  df-rest 16799  df-topgen 16820  df-top 21645  df-topon 21662  df-bases 21697  df-cld 21770  df-conn 22163
This theorem is referenced by: (None)
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