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Theorem consuba 17488
Description: Connectedness for a subspace. See connsub 17489. (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 17230 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2 dfcon2 17487 . . 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 2961 . . . . 5  |-  x  e. 
_V
54inex1 4347 . . . 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 16998 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
87adantr 453 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  X  e.  J )
9 simpr 449 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  C_  X )
108, 9ssexd 4353 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
11 elrest 13660 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
1210, 11syldan 458 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
13 vex 2961 . . . . . 6  |-  y  e. 
_V
1413inex1 4347 . . . . 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 13660 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1710, 16syldan 458 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1817adantr 453 . . . 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 733 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  u  =  ( x  i^i 
A ) )
2019neeq1d 2616 . . . . . 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 449 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  v  =  ( y  i^i 
A ) )
2221neeq1d 2616 . . . . . 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 3545 . . . . . . . 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 3561 . . . . . . . 8  |-  ( ( x  i^i  y )  i^i  A )  =  ( ( x  i^i 
A )  i^i  (
y  i^i  A )
)
2523, 24syl6eqr 2488 . . . . . . 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 2446 . . . . . 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 1255 . . . . 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 3504 . . . . . . 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 3591 . . . . . . 7  |-  ( ( x  u.  y )  i^i  A )  =  ( ( x  i^i 
A )  u.  (
y  i^i  A )
)
3028, 29syl6eqr 2488 . . . . . 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 2616 . . . . 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 313 . . . 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 4742 . . 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 4742 . 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 246 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708   _Vcvv 2958    u. cun 3320    i^i cin 3321    C_ wss 3322   (/)c0 3630   ` cfv 5457  (class class class)co 6084   ↾t crest 13653  TopOnctopon 16964   Conccon 17479
This theorem is referenced by:  connsub  17489  nconsubb  17491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-recs 6636  df-rdg 6671  df-oadd 6731  df-er 6908  df-en 7113  df-fin 7116  df-fi 7419  df-rest 13655  df-topgen 13672  df-top 16968  df-bases 16970  df-topon 16971  df-cld 17088  df-con 17480
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