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Theorem connsuba 22794
Description: Connectedness for a subspace. See connsub 22795. (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 22535 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝐽 β†Ύt 𝐴) ∈ (TopOnβ€˜π΄))
2 dfconn2 22793 . . 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 3451 . . . . 5 π‘₯ ∈ V
54inex1 5278 . . . 4 (π‘₯ ∩ 𝐴) ∈ V
65a1i 11 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ π‘₯ ∈ 𝐽) β†’ (π‘₯ ∩ 𝐴) ∈ V)
7 toponmax 22298 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
87adantr 482 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝑋 ∈ 𝐽)
9 simpr 486 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 βŠ† 𝑋)
108, 9ssexd 5285 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝐴 ∈ V)
11 elrest 17317 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ V) β†’ (𝑒 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑒 = (π‘₯ ∩ 𝐴)))
1210, 11syldan 592 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑒 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘₯ ∈ 𝐽 𝑒 = (π‘₯ ∩ 𝐴)))
13 vex 3451 . . . . . 6 𝑦 ∈ V
1413inex1 5278 . . . . 5 (𝑦 ∩ 𝐴) ∈ V
1514a1i 11 . . . 4 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑦 ∈ 𝐽) β†’ (𝑦 ∩ 𝐴) ∈ V)
16 elrest 17317 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 ∈ V) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
1710, 16syldan 592 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
1817adantr 482 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) β†’ (𝑣 ∈ (𝐽 β†Ύt 𝐴) ↔ βˆƒπ‘¦ ∈ 𝐽 𝑣 = (𝑦 ∩ 𝐴)))
19 simplr 768 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ 𝑒 = (π‘₯ ∩ 𝐴))
2019neeq1d 3000 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 β‰  βˆ… ↔ (π‘₯ ∩ 𝐴) β‰  βˆ…))
21 simpr 486 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ 𝑣 = (𝑦 ∩ 𝐴))
2221neeq1d 3000 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑣 β‰  βˆ… ↔ (𝑦 ∩ 𝐴) β‰  βˆ…))
2319, 21ineq12d 4177 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 ∩ 𝑣) = ((π‘₯ ∩ 𝐴) ∩ (𝑦 ∩ 𝐴)))
24 inindir 4191 . . . . . . . 8 ((π‘₯ ∩ 𝑦) ∩ 𝐴) = ((π‘₯ ∩ 𝐴) ∩ (𝑦 ∩ 𝐴))
2523, 24eqtr4di 2791 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 ∩ 𝑣) = ((π‘₯ ∩ 𝑦) ∩ 𝐴))
2625eqeq1d 2735 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ ((𝑒 ∩ 𝑣) = βˆ… ↔ ((π‘₯ ∩ 𝑦) ∩ 𝐴) = βˆ…))
2720, 22, 263anbi123d 1437 . . . . 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 4128 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 βˆͺ 𝑣) = ((π‘₯ ∩ 𝐴) βˆͺ (𝑦 ∩ 𝐴)))
29 indir 4239 . . . . . . 7 ((π‘₯ βˆͺ 𝑦) ∩ 𝐴) = ((π‘₯ ∩ 𝐴) βˆͺ (𝑦 ∩ 𝐴))
3028, 29eqtr4di 2791 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) ∧ 𝑒 = (π‘₯ ∩ 𝐴)) ∧ 𝑣 = (𝑦 ∩ 𝐴)) β†’ (𝑒 βˆͺ 𝑣) = ((π‘₯ βˆͺ 𝑦) ∩ 𝐴))
3130neeq1d 3000 . . . . 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 345 . . . 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 5369 . . 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 5369 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  β€˜cfv 6500  (class class class)co 7361   β†Ύt crest 17310  TopOnctopon 22282  Conncconn 22785
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-en 8890  df-fin 8893  df-fi 9355  df-rest 17312  df-topgen 17333  df-top 22266  df-topon 22283  df-bases 22319  df-cld 22393  df-conn 22786
This theorem is referenced by:  connsub  22795  nconnsubb  22797
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