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Theorem consuba 17146
Description: Connectedness for a subspace. See connsub 17147. (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 16892 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  ( Jt  A )  e.  (TopOn `  A ) )
2 dfcon2 17145 . . 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 15 . 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 2791 . . . . 5  |-  x  e. 
_V
54inex1 4155 . . . 4  |-  ( x  i^i  A )  e. 
_V
65a1i 10 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  x  e.  J )  ->  (
x  i^i  A )  e.  _V )
7 simpr 447 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  C_  X )
8 toponmax 16666 . . . . . 6  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
98adantr 451 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  X  e.  J )
10 ssexg 4160 . . . . 5  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
117, 9, 10syl2anc 642 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  A  e.  _V )
12 elrest 13332 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
1311, 12syldan 456 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
u  e.  ( Jt  A )  <->  E. x  e.  J  u  =  ( x  i^i  A ) ) )
14 vex 2791 . . . . . 6  |-  y  e. 
_V
1514inex1 4155 . . . . 5  |-  ( y  i^i  A )  e. 
_V
1615a1i 10 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  y  e.  J )  ->  (
y  i^i  A )  e.  _V )
17 elrest 13332 . . . . . 6  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  _V )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1811, 17syldan 456 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  ->  (
v  e.  ( Jt  A )  <->  E. y  e.  J  v  =  ( y  i^i  A ) ) )
1918adantr 451 . . . 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
) ) )
20 simplr 731 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  u  =  ( x  i^i 
A ) )
2120neeq1d 2459 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
u  =/=  (/)  <->  ( x  i^i  A )  =/=  (/) ) )
22 simpr 447 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  v  =  ( y  i^i 
A ) )
2322neeq1d 2459 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  A  C_  X )  /\  u  =  ( x  i^i  A ) )  /\  v  =  ( y  i^i  A
) )  ->  (
v  =/=  (/)  <->  ( y  i^i  A )  =/=  (/) ) )
2420, 22ineq12d 3371 . . . . . . . 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
) ) )
25 inindir 3387 . . . . . . . 8  |-  ( ( x  i^i  y )  i^i  A )  =  ( ( x  i^i 
A )  i^i  (
y  i^i  A )
)
2624, 25syl6eqr 2333 . . . . . . 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 ) )
2726eqeq1d 2291 . . . . . 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 )  =  (/) ) )
2821, 23, 273anbi123d 1252 . . . . 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 )  =  (/) ) ) )
2920, 22uneq12d 3330 . . . . . . 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
) ) )
30 indir 3417 . . . . . . 7  |-  ( ( x  u.  y )  i^i  A )  =  ( ( x  i^i 
A )  u.  (
y  i^i  A )
)
3129, 30syl6eqr 2333 . . . . . 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 ) )
3231neeq1d 2459 . . . . 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 ) )
3328, 32imbi12d 311 . . . 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 ) ) )
3416, 19, 33ralxfr2d 4550 . . 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 ) ) )
356, 13, 34ralxfr2d 4550 . 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 ) ) )
363, 35bitrd 244 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   _Vcvv 2788    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   ` cfv 5255  (class class class)co 5858   ↾t crest 13325  TopOnctopon 16632   Conccon 17137
This theorem is referenced by:  connsub  17147  nconsubb  17149
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-oadd 6483  df-er 6660  df-en 6864  df-fin 6867  df-fi 7165  df-rest 13327  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cld 16756  df-con 17138
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