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Theorem consuba 19029
Description: Connectedness for a subspace. See connsub 19030. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
Assertion
Ref Expression
consuba  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
( Jt  A )  e.  Con  <->  A. x  e.  J  A. y  e.  J  (
( ( x  i^i 
A )  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  ->  ( ( x  u.  y )  i^i  A )  =/= 
A ) ) )
Distinct variable groups:    x, y, A    x, J, y    x, X, y

Proof of Theorem consuba
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resttopon 18770 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2 dfcon2 19028 . . 3  |-  ( ( Jt  A )  e.  (TopOn `  A )  ->  (
( Jt  A )  e.  Con  <->  A. u  e.  ( Jt  A
) A. v  e.  ( Jt  A ) ( ( u  =/=  (/)  /\  v  =/=  (/)  /\  ( u  i^i  v )  =  (/) )  ->  ( u  u.  v )  =/= 
A ) ) )
31, 2syl 16 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
( Jt  A )  e.  Con  <->  A. u  e.  ( Jt  A
) A. v  e.  ( Jt  A ) ( ( u  =/=  (/)  /\  v  =/=  (/)  /\  ( u  i^i  v )  =  (/) )  ->  ( u  u.  v )  =/= 
A ) ) )
4 vex 2980 . . . . 5  |-  x  e. 
_V
54inex1 4438 . . . 4  |-  ( x  i^i  A )  e. 
_V
65a1i 11 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  x  e.  J )  ->  (
x  i^i  A )  e.  _V )
7 toponmax 18538 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
87adantr 465 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  X  e.  J )
9 simpr 461 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  C_  X )
108, 9ssexd 4444 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
11 elrest 14371 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
1210, 11syldan 470 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
13 vex 2980 . . . . . 6  |-  y  e. 
_V
1413inex1 4438 . . . . 5  |-  ( y  i^i  A )  e. 
_V
1514a1i 11 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  y  e.  J )  ->  (
y  i^i  A )  e.  _V )
16 elrest 14371 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1710, 16syldan 470 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1817adantr 465 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i 
A ) )  -> 
( v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A
) ) )
19 simplr 754 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  u  =  ( x  i^i 
A ) )
2019neeq1d 2626 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
u  =/=  (/)  <->  ( x  i^i  A )  =/=  (/) ) )
21 simpr 461 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  v  =  ( y  i^i 
A ) )
2221neeq1d 2626 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
v  =/=  (/)  <->  ( y  i^i  A )  =/=  (/) ) )
2319, 21ineq12d 3558 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
u  i^i  v )  =  ( ( x  i^i  A )  i^i  ( y  i^i  A
) ) )
24 inindir 3573 . . . . . . . 8  |-  ( ( x  i^i  y )  i^i  A )  =  ( ( x  i^i 
A )  i^i  (
y  i^i  A )
)
2523, 24syl6eqr 2493 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
u  i^i  v )  =  ( ( x  i^i  y )  i^i 
A ) )
2625eqeq1d 2451 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
( u  i^i  v
)  =  (/)  <->  ( (
x  i^i  y )  i^i  A )  =  (/) ) )
2720, 22, 263anbi123d 1289 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
( u  =/=  (/)  /\  v  =/=  (/)  /\  ( u  i^i  v )  =  (/) )  <->  ( ( x  i^i  A )  =/=  (/)  /\  ( y  i^i 
A )  =/=  (/)  /\  (
( x  i^i  y
)  i^i  A )  =  (/) ) ) )
2819, 21uneq12d 3516 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
u  u.  v )  =  ( ( x  i^i  A )  u.  ( y  i^i  A
) ) )
29 indir 3603 . . . . . . 7  |-  ( ( x  u.  y )  i^i  A )  =  ( ( x  i^i 
A )  u.  (
y  i^i  A )
)
3028, 29syl6eqr 2493 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
u  u.  v )  =  ( ( x  u.  y )  i^i 
A ) )
3130neeq1d 2626 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
( u  u.  v
)  =/=  A  <->  ( (
x  u.  y )  i^i  A )  =/= 
A ) )
3227, 31imbi12d 320 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
( ( u  =/=  (/)  /\  v  =/=  (/)  /\  (
u  i^i  v )  =  (/) )  ->  (
u  u.  v )  =/=  A )  <->  ( (
( x  i^i  A
)  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  ->  ( ( x  u.  y )  i^i  A )  =/= 
A ) ) )
3315, 18, 32ralxfr2d 4513 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i 
A ) )  -> 
( A. v  e.  ( Jt  A ) ( ( u  =/=  (/)  /\  v  =/=  (/)  /\  ( u  i^i  v )  =  (/) )  ->  ( u  u.  v )  =/= 
A )  <->  A. y  e.  J  ( (
( x  i^i  A
)  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  ->  ( ( x  u.  y )  i^i  A )  =/= 
A ) ) )
346, 12, 33ralxfr2d 4513 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( A. u  e.  ( Jt  A ) A. v  e.  ( Jt  A ) ( ( u  =/=  (/)  /\  v  =/=  (/)  /\  ( u  i^i  v )  =  (/) )  ->  ( u  u.  v )  =/= 
A )  <->  A. x  e.  J  A. y  e.  J  ( (
( x  i^i  A
)  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  ->  ( ( x  u.  y )  i^i  A )  =/= 
A ) ) )
353, 34bitrd 253 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
( Jt  A )  e.  Con  <->  A. x  e.  J  A. y  e.  J  (
( ( x  i^i 
A )  =/=  (/)  /\  (
y  i^i  A )  =/=  (/)  /\  ( ( x  i^i  y )  i^i  A )  =  (/) )  ->  ( ( x  u.  y )  i^i  A )  =/= 
A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   E.wrex 2721   _Vcvv 2977    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   ` cfv 5423  (class class class)co 6096   ↾t crest 14364  TopOnctopon 18504   Conccon 19020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-oadd 6929  df-er 7106  df-en 7316  df-fin 7319  df-fi 7666  df-rest 14366  df-topgen 14387  df-top 18508  df-bases 18510  df-topon 18511  df-cld 18628  df-con 19021
This theorem is referenced by:  connsub  19030  nconsubb  19032
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