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Theorem consuba 19715
Description: Connectedness for a subspace. See connsub 19716. (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 19456 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2 dfcon2 19714 . . 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 3116 . . . . 5  |-  x  e. 
_V
54inex1 4588 . . . 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 19224 . . . . . 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 4594 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
11 elrest 14683 . . . 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 3116 . . . . . 6  |-  y  e. 
_V
1413inex1 4588 . . . . 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 14683 . . . . . 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 2744 . . . . . 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 2744 . . . . . 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 3701 . . . . . . . 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 3716 . . . . . . . 8  |-  ( ( x  i^i  y )  i^i  A )  =  ( ( x  i^i 
A )  i^i  (
y  i^i  A )
)
2523, 24syl6eqr 2526 . . . . . . 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 2469 . . . . . 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 1299 . . . . 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 3659 . . . . . . 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 3746 . . . . . . 7  |-  ( ( x  u.  y )  i^i  A )  =  ( ( x  i^i 
A )  u.  (
y  i^i  A )
)
3028, 29syl6eqr 2526 . . . . . 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 2744 . . . . 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 4663 . . 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 4663 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   _Vcvv 3113    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ` cfv 5588  (class class class)co 6284   ↾t crest 14676  TopOnctopon 19190   Conccon 19706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-oadd 7134  df-er 7311  df-en 7517  df-fin 7520  df-fi 7871  df-rest 14678  df-topgen 14699  df-top 19194  df-bases 19196  df-topon 19197  df-cld 19314  df-con 19707
This theorem is referenced by:  connsub  19716  nconsubb  19718
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