Theorem iunconnALT 44932

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 44932 verifies https://us.metamath.org/other/completeusersproof/iunconaltvd.html 44932. (Contributed by Alan Sare, 22-Apr-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
iunconnALT.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
iunconnALT.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
iunconnALT.3	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
iunconnALT.4	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)

Assertion

Ref	Expression
iunconnALT	⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		biid 261	. 2 ⊢ (((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑢 ∩ ∪ 𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑣 ∩ ∪ 𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑢 ∩ 𝑣) ⊆ (𝑋 ∖ ∪ 𝑘 ∈ 𝐴 𝐵)) ∧ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ (𝑢 ∪ 𝑣)) ↔ ((((((𝜑 ∧ 𝑢 ∈ 𝐽) ∧ 𝑣 ∈ 𝐽) ∧ (𝑢 ∩ ∪ 𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑣 ∩ ∪ 𝑘 ∈ 𝐴 𝐵) ≠ ∅) ∧ (𝑢 ∩ 𝑣) ⊆ (𝑋 ∖ ∪ 𝑘 ∈ 𝐴 𝐵)) ∧ ∪ 𝑘 ∈ 𝐴 𝐵 ⊆ (𝑢 ∪ 𝑣)))
2		iunconnALT.1	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		iunconnALT.2	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ⊆ 𝑋)
4		iunconnALT.3	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐵)
5		iunconnALT.4	. 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐽 ↾t 𝐵) ∈ Conn)
6	1, 2, 3, 4, 5	iunconnlem2 44931	1 ⊢ (𝜑 → (𝐽 ↾t ∪ 𝑘 ∈ 𝐴 𝐵) ∈ Conn)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109   ≠ wne 2926   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  ∪ ciun 4958  ‘cfv 6514  (class class class)co 7390   ↾_t crest 17390  TopOnctopon 22804  Conncconn 23305

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-en 8922  df-fin 8925  df-fi 9369  df-rest 17392  df-topgen 17413  df-top 22788  df-topon 22805  df-bases 22840  df-cld 22913  df-conn 23306

This theorem is referenced by: (None)