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Theorem connsuba 23344
Description: Connectedness for a subspace. See connsub 23345. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
Assertion
Ref Expression
connsuba ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝐴) ∈ Conn ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ 𝐽 (((π‘₯ ∩ 𝐴) β‰  βˆ… ∧ (𝑦 ∩ 𝐴) β‰  βˆ… ∧ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…) β†’ ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) β‰  𝐴)))
Distinct variable groups:   π‘₯,𝑦,𝐴   π‘₯,𝐽,𝑦   π‘₯,𝑋,𝑦

Proof of Theorem connsuba
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resttopon 23085 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
2 dfconn2 23343 . . 3 ((𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄) β†’ ((𝐽 β†Ύt 𝐴) ∈ Conn ↔ βˆ€π‘’ ∈ (𝐽 β†Ύt 𝐴)βˆ€π‘£ ∈ (𝐽 β†Ύt 𝐴)((𝑒 β‰  βˆ… ∧ 𝑣 β‰  βˆ… ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (𝑒 βˆͺ 𝑣) β‰  𝐴)))
31, 2syl 17 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝐴) ∈ Conn ↔ βˆ€π‘’ ∈ (𝐽 β†Ύt 𝐴)βˆ€π‘£ ∈ (𝐽 β†Ύt 𝐴)((𝑒 β‰  βˆ… ∧ 𝑣 β‰  βˆ… ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (𝑒 βˆͺ 𝑣) β‰  𝐴)))
4 vex 3477 . . . . 5 π‘₯ ∈ V
54inex1 5321 . . . 4 (π‘₯ ∩ 𝐴) ∈ V
65a1i 11 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ ∩ 𝐴) ∈ V)
7 toponmax 22848 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
87adantr 479 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝑋 ∈ 𝐽)
9 simpr 483 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 βŠ† 𝑋)
108, 9ssexd 5328 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 ∈ V)
11 elrest 17416 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ V) β†’ (𝑒 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑒 = (π‘₯ ∩ 𝐴)))
1210, 11syldan 589 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑒 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑒 = (π‘₯ ∩ 𝐴)))
13 vex 3477 . . . . . 6 𝑦 ∈ V
1413inex1 5321 . . . . 5 (𝑦 ∩ 𝐴) ∈ V
1514a1i 11 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) β†’ (𝑦 ∩ 𝐴) ∈ V)
16 elrest 17416 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ V) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
1710, 16syldan 589 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
1817adantr 479 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
19 simplr 767 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ 𝑒 = (π‘₯ ∩ 𝐴))
2019neeq1d 2997 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 β‰  βˆ… ↔ (π‘₯ ∩ 𝐴) β‰  βˆ…))
21 simpr 483 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ 𝑣 = (𝑦 ∩ 𝐴))
2221neeq1d 2997 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑣 β‰  βˆ… ↔ (𝑦 ∩ 𝐴) β‰  βˆ…))
2319, 21ineq12d 4215 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 ∩ 𝑣) = ((π‘₯ ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24 inindir 4230 . . . . . . . 8 ((π‘₯ ∩ 𝑦) ∩ 𝐴) = ((π‘₯ ∩ 𝐴) ∩ (𝑦 ∩ 𝐴))
2523, 24eqtr4di 2786 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 ∩ 𝑣) = ((π‘₯ ∩ 𝑦) ∩ 𝐴))
2625eqeq1d 2730 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ ((𝑒 ∩ 𝑣) = βˆ… ↔ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…))
2720, 22, 263anbi123d 1432 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ ((𝑒 β‰  βˆ… ∧ 𝑣 β‰  βˆ… ∧ (𝑒 ∩ 𝑣) = βˆ…) ↔ ((π‘₯ ∩ 𝐴) β‰  βˆ… ∧ (𝑦 ∩ 𝐴) β‰  βˆ… ∧ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…)))
2819, 21uneq12d 4165 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 βˆͺ 𝑣) = ((π‘₯ ∩ 𝐴) βˆͺ (𝑦 ∩ 𝐴)))
29 indir 4278 . . . . . . 7 ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) = ((π‘₯ ∩ 𝐴) βˆͺ (𝑦 ∩ 𝐴))
3028, 29eqtr4di 2786 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 βˆͺ 𝑣) = ((π‘₯ βˆͺ 𝑦) ∩ 𝐴))
3130neeq1d 2997 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ ((𝑒 βˆͺ 𝑣) β‰  𝐴 ↔ ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) β‰  𝐴))
3227, 31imbi12d 343 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (((𝑒 β‰  βˆ… ∧ 𝑣 β‰  βˆ… ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (𝑒 βˆͺ 𝑣) β‰  𝐴) ↔ (((π‘₯ ∩ 𝐴) β‰  βˆ… ∧ (𝑦 ∩ 𝐴) β‰  βˆ… ∧ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…) β†’ ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) β‰  𝐴)))
3315, 18, 32ralxfr2d 5414 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) β†’ (βˆ€π‘£ ∈ (𝐽 β†Ύt 𝐴)((𝑒 β‰  βˆ… ∧ 𝑣 β‰  βˆ… ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (𝑒 βˆͺ 𝑣) β‰  𝐴) ↔ βˆ€π‘¦ ∈ 𝐽 (((π‘₯ ∩ 𝐴) β‰  βˆ… ∧ (𝑦 ∩ 𝐴) β‰  βˆ… ∧ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…) β†’ ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) β‰  𝐴)))
346, 12, 33ralxfr2d 5414 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (βˆ€π‘’ ∈ (𝐽 β†Ύt 𝐴)βˆ€π‘£ ∈ (𝐽 β†Ύt 𝐴)((𝑒 β‰  βˆ… ∧ 𝑣 β‰  βˆ… ∧ (𝑒 ∩ 𝑣) = βˆ…) β†’ (𝑒 βˆͺ 𝑣) β‰  𝐴) ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ 𝐽 (((π‘₯ ∩ 𝐴) β‰  βˆ… ∧ (𝑦 ∩ 𝐴) β‰  βˆ… ∧ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…) β†’ ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) β‰  𝐴)))
353, 34bitrd 278 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ ((𝐽 β†Ύt 𝐴) ∈ Conn ↔ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ 𝐽 (((π‘₯ ∩ 𝐴) β‰  βˆ… ∧ (𝑦 ∩ 𝐴) β‰  βˆ… ∧ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…) β†’ ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) β‰  𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3473   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4326  β€˜cfv 6553  (class class class)co 7426   β†Ύt crest 17409  TopOnctopon 22832  Conncconn 23335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-en 8971  df-fin 8974  df-fi 9442  df-rest 17411  df-topgen 17432  df-top 22816  df-topon 22833  df-bases 22869  df-cld 22943  df-conn 23336
This theorem is referenced by:  connsub  23345  nconnsubb  23347
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