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Theorem consuba 17444
Description: Connectedness for a subspace. See connsub 17445. (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 17187 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2 dfcon2 17443 . . 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 2927 . . . . 5  |-  x  e. 
_V
54inex1 4312 . . . 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 16956 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
87adantr 452 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  X  e.  J )
9 simpr 448 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  C_  X )
108, 9ssexd 4318 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
11 elrest 13618 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
1210, 11syldan 457 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
13 vex 2927 . . . . . 6  |-  y  e. 
_V
1413inex1 4312 . . . . 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 13618 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1710, 16syldan 457 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1817adantr 452 . . . 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 732 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  u  =  ( x  i^i 
A ) )
2019neeq1d 2588 . . . . . 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 448 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  v  =  ( y  i^i 
A ) )
2221neeq1d 2588 . . . . . 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 3511 . . . . . . . 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 3527 . . . . . . . 8  |-  ( ( x  i^i  y )  i^i  A )  =  ( ( x  i^i 
A )  i^i  (
y  i^i  A )
)
2523, 24syl6eqr 2462 . . . . . . 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 2420 . . . . . 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 1254 . . . . 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 3470 . . . . . . 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 3557 . . . . . . 7  |-  ( ( x  u.  y )  i^i  A )  =  ( ( x  i^i 
A )  u.  (
y  i^i  A )
)
3028, 29syl6eqr 2462 . . . . . 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 2588 . . . . 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 312 . . . 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 4706 . . 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 4706 . 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 245 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   _Vcvv 2924    u. cun 3286    i^i cin 3287    C_ wss 3288   (/)c0 3596   ` cfv 5421  (class class class)co 6048   ↾t crest 13611  TopOnctopon 16922   Conccon 17435
This theorem is referenced by:  connsub  17445  nconsubb  17447
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-recs 6600  df-rdg 6635  df-oadd 6695  df-er 6872  df-en 7077  df-fin 7080  df-fi 7382  df-rest 13613  df-topgen 13630  df-top 16926  df-bases 16928  df-topon 16929  df-cld 17046  df-con 17436
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