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Theorem unconn 23153
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 4346 . . 3 ((𝐴 ∩ 𝐡) β‰  βˆ… ↔ βˆƒπ‘₯ π‘₯ ∈ (𝐴 ∩ 𝐡))
2 uniiun 5061 . . . . . . . . 9 βˆͺ {𝐴, 𝐡} = βˆͺ π‘˜ ∈ {𝐴, 𝐡}π‘˜
3 simpl1 1191 . . . . . . . . . . . 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 22648 . . . . . . . . . . . 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 1226 . . . . . . . . . . 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 5324 . . . . . . . . . 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 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†’ 𝐡 βŠ† 𝑋)
95, 8ssexd 5324 . . . . . . . . . 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 4925 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝐡 ∈ V) β†’ βˆͺ {𝐴, 𝐡} = (𝐴 βˆͺ 𝐡))
117, 9, 10syl2anc 584 . . . . . . . . 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 2786 . . . . . . . 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 7427 . . . . . . 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 3478 . . . . . . . . . 10 π‘˜ ∈ V
1514elpr 4651 . . . . . . . . 9 (π‘˜ ∈ {𝐴, 𝐡} ↔ (π‘˜ = 𝐴 ∨ π‘˜ = 𝐡))
16 simpl2 1192 . . . . . . . . . 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 4007 . . . . . . . . . . . 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 4007 . . . . . . . . . . . 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 954 . . . . . . . . . 10 ((π‘˜ = 𝐴 ∨ π‘˜ = 𝐡) β†’ ((𝐴 βŠ† 𝑋 ∧ 𝐡 βŠ† 𝑋) β†’ π‘˜ βŠ† 𝑋))
2216, 21mpan9 507 . . . . . . . . 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 594 . . . . . . . 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 1193 . . . . . . . . . . 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 3964 . . . . . . . . . . 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 2822 . . . . . . . . . . . 12 (π‘˜ = 𝐴 β†’ (π‘₯ ∈ π‘˜ ↔ π‘₯ ∈ 𝐴))
2827biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐴 β†’ (π‘₯ ∈ 𝐴 β†’ π‘₯ ∈ π‘˜))
29 eleq2 2822 . . . . . . . . . . . 12 (π‘˜ = 𝐡 β†’ (π‘₯ ∈ π‘˜ ↔ π‘₯ ∈ 𝐡))
3029biimprd 247 . . . . . . . . . . 11 (π‘˜ = 𝐡 β†’ (π‘₯ ∈ 𝐡 β†’ π‘₯ ∈ π‘˜))
3128, 30jaoa 954 . . . . . . . . . 10 ((π‘˜ = 𝐴 ∨ π‘˜ = 𝐡) β†’ ((π‘₯ ∈ 𝐴 ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ π‘˜))
3226, 31mpan9 507 . . . . . . . . 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 594 . . . . . . . 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 485 . . . . . . . . . 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 7419 . . . . . . . . . . . . 13 (π‘˜ = 𝐴 β†’ (𝐽 β†Ύt π‘˜) = (𝐽 β†Ύt 𝐴))
3635eleq1d 2818 . . . . . . . . . . . 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 7419 . . . . . . . . . . . . 13 (π‘˜ = 𝐡 β†’ (𝐽 β†Ύt π‘˜) = (𝐽 β†Ύt 𝐡))
3938eleq1d 2818 . . . . . . . . . . . 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 954 . . . . . . . . . 10 ((π‘˜ = 𝐴 ∨ π‘˜ = 𝐡) β†’ (((𝐽 β†Ύt 𝐴) ∈ Conn ∧ (𝐽 β†Ύt 𝐡) ∈ Conn) β†’ (𝐽 β†Ύt π‘˜) ∈ Conn))
4234, 41mpan9 507 . . . . . . . . 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 594 . . . . . . . 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 23152 . . . . . . 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 2834 . . . . . 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 413 . . . . 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 1121 . . . 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 1936 . . 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 1117 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 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940  Vcvv 3474   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  {cpr 4630  βˆͺ cuni 4908  βˆͺ ciun 4997  β€˜cfv 6543  (class class class)co 7411   β†Ύt crest 17370  TopOnctopon 22632  Conncconn 23135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-cld 22743  df-conn 23136
This theorem is referenced by: (None)
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