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Theorem unconn 22933
Description: The union of two connected overlapping subspaces is connected. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 11-Jun-2014.)
Assertion
Ref Expression
unconn ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ (𝐴 ∩ 𝐡) β‰  βˆ…) β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)) ∈ Conn))
Proof of Theorem unconn
Dummy variables π‘₯ π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4347 . . 3 ((𝐴 ∩ 𝐡) β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ (𝐴 ∩ 𝐡))
2 uniiun 5062 . . . . . . . . 9 βˆͺ {𝐴, 𝐡} = βˆͺ π‘˜ ∈ {𝐴, 𝐡}π‘˜
3 simpl1 1192 . . . . . . . . . . . 12 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
4 toponmax 22428 . . . . . . . . . . . 12 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
53, 4syl 17 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ 𝑋 ∈ 𝐽)
6 simpl2l 1227 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ 𝐴 βŠ† 𝑋)
75, 6ssexd 5325 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ 𝐴 ∈ V)
8 simpl2r 1228 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ 𝐡 βŠ† 𝑋)
95, 8ssexd 5325 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ 𝐡 ∈ V)
10 uniprg 4926 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐡 ∈ V) β†’ βˆͺ {𝐴, 𝐡} = (𝐴 βˆͺ 𝐡))
117, 9, 10syl2anc 585 . . . . . . . . 9 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ βˆͺ {𝐴, 𝐡} = (𝐴 βˆͺ 𝐡))
122, 11eqtr3id 2787 . . . . . . . 8 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ βˆͺ π‘˜ ∈ {𝐴, 𝐡}π‘˜ = (𝐴 βˆͺ 𝐡))
1312oveq2d 7425 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ (𝐽 β†Ύt βˆͺ π‘˜ ∈ {𝐴, 𝐡}π‘˜) = (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)))
14 vex 3479 . . . . . . . . . 10 π‘˜ ∈ V
1514elpr 4652 . . . . . . . . 9 (π‘˜ ∈ {𝐴, 𝐡} ↔ (π‘˜ = 𝐴 ∨ π‘˜ = 𝐡))
16 simpl2 1193 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋))
17 sseq1 4008 . . . . . . . . . . . 12 (π‘˜ = 𝐴 β†’ (π‘˜ βŠ† 𝑋 ↔ 𝐴 βŠ† 𝑋))
1817biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐴 β†’ (𝐴 βŠ† 𝑋 β†’ π‘˜ βŠ† 𝑋))
19 sseq1 4008 . . . . . . . . . . . 12 (π‘˜ = 𝐡 β†’ (π‘˜ βŠ† 𝑋 ↔ 𝐡 βŠ† 𝑋))
2019biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐡 β†’ (𝐡 βŠ† 𝑋 β†’ π‘˜ βŠ† 𝑋))
2118, 20jaoa 955 . . . . . . . . . 10 ((π‘˜ = 𝐴 ∨ π‘˜ = 𝐡) β†’ ((𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ π‘˜ βŠ† 𝑋))
2216, 21mpan9 508 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) ∧ (π‘˜ = 𝐴 ∨ π‘˜ = 𝐡)) β†’ π‘˜ βŠ† 𝑋)
2315, 22sylan2b 595 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) ∧ π‘˜ ∈ {𝐴, 𝐡}) β†’ π‘˜ βŠ† 𝑋)
24 simpl3 1194 . . . . . . . . . . 11 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ π‘₯ ∈ (𝐴 ∩ 𝐡))
25 elin 3965 . . . . . . . . . . 11 (π‘₯ ∈ (𝐴 ∩ 𝐡) ↔ (π‘₯ ∈ 𝐴 ∧ π‘₯ ∈ 𝐡))
2624, 25sylib 217 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ (π‘₯ ∈ 𝐴 ∧ π‘₯ ∈ 𝐡))
27 eleq2 2823 . . . . . . . . . . . 12 (π‘˜ = 𝐴 β†’ (π‘₯ ∈ π‘˜ ↔ π‘₯ ∈ 𝐴))
2827biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ π‘˜))
29 eleq2 2823 . . . . . . . . . . . 12 (π‘˜ = 𝐡 β†’ (π‘₯ ∈ π‘˜ ↔ π‘₯ ∈ 𝐡))
3029biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐡 β†’ (π‘₯ ∈ 𝐡 β†’ π‘₯ ∈ π‘˜))
3128, 30jaoa 955 . . . . . . . . . 10 ((π‘˜ = 𝐴 ∨ π‘˜ = 𝐡) β†’ ((π‘₯ ∈ 𝐴 ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ π‘˜))
3226, 31mpan9 508 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) ∧ (π‘˜ = 𝐴 ∨ π‘˜ = 𝐡)) β†’ π‘₯ ∈ π‘˜)
3315, 32sylan2b 595 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) ∧ π‘˜ ∈ {𝐴, 𝐡}) β†’ π‘₯ ∈ π‘˜)
34 simpr 486 . . . . . . . . . 10 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn))
35 oveq2 7417 . . . . . . . . . . . . 13 (π‘˜ = 𝐴 β†’ (𝐽 β†Ύt π‘˜) = (𝐽 β†Ύt 𝐴))
3635eleq1d 2819 . . . . . . . . . . . 12 (π‘˜ = 𝐴 β†’ ((𝐽 β†Ύt π‘˜) ∈ Conn ↔ (𝐽 β†Ύt 𝐴) ∈ Conn))
3736biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐴 β†’ ((𝐽 β†Ύt 𝐴) ∈ Conn β†’ (𝐽 β†Ύt π‘˜) ∈ Conn))
38 oveq2 7417 . . . . . . . . . . . . 13 (π‘˜ = 𝐡 β†’ (𝐽 β†Ύt π‘˜) = (𝐽 β†Ύt 𝐡))
3938eleq1d 2819 . . . . . . . . . . . 12 (π‘˜ = 𝐡 β†’ ((𝐽 β†Ύt π‘˜) ∈ Conn ↔ (𝐽 β†Ύt 𝐡) ∈ Conn))
4039biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐡 β†’ ((𝐽 β†Ύt 𝐡) ∈ Conn β†’ (𝐽 β†Ύt π‘˜) ∈ Conn))
4137, 40jaoa 955 . . . . . . . . . 10 ((π‘˜ = 𝐴 ∨ π‘˜ = 𝐡) β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt π‘˜) ∈ Conn))
4234, 41mpan9 508 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) ∧ (π‘˜ = 𝐴 ∨ π‘˜ = 𝐡)) β†’ (𝐽 β†Ύt π‘˜) ∈ Conn)
4315, 42sylan2b 595 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) ∧ π‘˜ ∈ {𝐴, 𝐡}) β†’ (𝐽 β†Ύt π‘˜) ∈ Conn)
443, 23, 33, 43iunconn 22932 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ (𝐽 β†Ύt βˆͺ π‘˜ ∈ {𝐴, 𝐡}π‘˜) ∈ Conn)
4513, 44eqeltrrd 2835 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) ∧ ((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn)) β†’ (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)) ∈ Conn)
4645ex 414 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ π‘₯ ∈ (𝐴 ∩ 𝐡)) β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)) ∈ Conn))
47463expia 1122 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋)) β†’ (π‘₯ ∈ (𝐴 ∩ 𝐡) β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)) ∈ Conn)))
4847exlimdv 1937 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋)) β†’ (βˆƒπ‘₯ π‘₯ ∈ (𝐴 ∩ 𝐡) β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)) ∈ Conn)))
491, 48biimtrid 241 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋)) β†’ ((𝐴 ∩ 𝐡) β‰  βˆ… β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)) ∈ Conn)))
50493impia 1118 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ (𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) ∧ (𝐴 ∩ 𝐡) β‰  βˆ…) β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt (𝐴 βˆͺ 𝐡)) ∈ Conn))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2941  Vcvv 3475   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {cpr 4631  βˆͺ cuni 4909  βˆͺ ciun 4998  β€˜cfv 6544  (class class class)co 7409   β†Ύt crest 17366  TopOnctopon 22412  Conncconn 22915
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-en 8940  df-fin 8943  df-fi 9406  df-rest 17368  df-topgen 17389  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-conn 22916
This theorem is referenced by: (None)
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