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Theorem connsuba 23274
Description: Connectedness for a subspace. See connsub 23275. (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 23015 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
2 dfconn2 23273 . . 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 3472 . . . . 5 π‘₯ ∈ V
54inex1 5310 . . . 4 (π‘₯ ∩ 𝐴) ∈ V
65a1i 11 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ ∩ 𝐴) ∈ V)
7 toponmax 22778 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
87adantr 480 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝑋 ∈ 𝐽)
9 simpr 484 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 βŠ† 𝑋)
108, 9ssexd 5317 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 ∈ V)
11 elrest 17379 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ V) β†’ (𝑒 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑒 = (π‘₯ ∩ 𝐴)))
1210, 11syldan 590 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑒 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑒 = (π‘₯ ∩ 𝐴)))
13 vex 3472 . . . . . 6 𝑦 ∈ V
1413inex1 5310 . . . . 5 (𝑦 ∩ 𝐴) ∈ V
1514a1i 11 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) β†’ (𝑦 ∩ 𝐴) ∈ V)
16 elrest 17379 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ V) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
1710, 16syldan 590 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
1817adantr 480 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
19 simplr 766 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ 𝑒 = (π‘₯ ∩ 𝐴))
2019neeq1d 2994 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 β‰  βˆ… ↔ (π‘₯ ∩ 𝐴) β‰  βˆ…))
21 simpr 484 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ 𝑣 = (𝑦 ∩ 𝐴))
2221neeq1d 2994 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑣 β‰  βˆ… ↔ (𝑦 ∩ 𝐴) β‰  βˆ…))
2319, 21ineq12d 4208 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 ∩ 𝑣) = ((π‘₯ ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24 inindir 4222 . . . . . . . 8 ((π‘₯ ∩ 𝑦) ∩ 𝐴) = ((π‘₯ ∩ 𝐴) ∩ (𝑦 ∩ 𝐴))
2523, 24eqtr4di 2784 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 ∩ 𝑣) = ((π‘₯ ∩ 𝑦) ∩ 𝐴))
2625eqeq1d 2728 . . . . . 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 4159 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 βˆͺ 𝑣) = ((π‘₯ ∩ 𝐴) βˆͺ (𝑦 ∩ 𝐴)))
29 indir 4270 . . . . . . 7 ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) = ((π‘₯ ∩ 𝐴) βˆͺ (𝑦 ∩ 𝐴))
3028, 29eqtr4di 2784 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 βˆͺ 𝑣) = ((π‘₯ βˆͺ 𝑦) ∩ 𝐴))
3130neeq1d 2994 . . . . 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 344 . . . 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 5401 . . 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 5401 . 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 279 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   βˆͺ cun 3941   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  β€˜cfv 6536  (class class class)co 7404   β†Ύt crest 17372  TopOnctopon 22762  Conncconn 23265
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-en 8939  df-fin 8942  df-fi 9405  df-rest 17374  df-topgen 17395  df-top 22746  df-topon 22763  df-bases 22799  df-cld 22873  df-conn 23266
This theorem is referenced by:  connsub  23275  nconnsubb  23277
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