MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qtopcn Unicode version

Theorem qtopcn 17405
Description: Universal property of a quotient map. (Contributed by Mario Carneiro, 23-Mar-2015.)
Assertion
Ref Expression
qtopcn  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( G  e.  ( ( J qTop  F )  Cn  K )  <->  ( G  o.  F )  e.  ( J  Cn  K ) ) )

Proof of Theorem qtopcn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simplll 734 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  J  e.  (TopOn `  X )
)
2 simplrl 736 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  F : X -onto-> Y )
3 elqtop3 17394 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y )  ->  (
( `' G "
x )  e.  ( J qTop  F )  <->  ( ( `' G " x ) 
C_  Y  /\  ( `' F " ( `' G " x ) )  e.  J ) ) )
41, 2, 3syl2anc 642 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  (
( `' G "
x )  e.  ( J qTop  F )  <->  ( ( `' G " x ) 
C_  Y  /\  ( `' F " ( `' G " x ) )  e.  J ) ) )
5 cnvimass 5033 . . . . . . . 8  |-  ( `' G " x ) 
C_  dom  G
6 simplrr 737 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  G : Y --> Z )
7 fdm 5393 . . . . . . . . 9  |-  ( G : Y --> Z  ->  dom  G  =  Y )
86, 7syl 15 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  dom  G  =  Y )
95, 8syl5sseq 3226 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  ( `' G " x ) 
C_  Y )
109biantrurd 494 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  (
( `' F "
( `' G "
x ) )  e.  J  <->  ( ( `' G " x ) 
C_  Y  /\  ( `' F " ( `' G " x ) )  e.  J ) ) )
114, 10bitr4d 247 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  (
( `' G "
x )  e.  ( J qTop  F )  <->  ( `' F " ( `' G " x ) )  e.  J ) )
12 cnvco 4865 . . . . . . . 8  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
1312imaeq1i 5009 . . . . . . 7  |-  ( `' ( G  o.  F
) " x )  =  ( ( `' F  o.  `' G
) " x )
14 imaco 5178 . . . . . . 7  |-  ( ( `' F  o.  `' G ) " x
)  =  ( `' F " ( `' G " x ) )
1513, 14eqtri 2303 . . . . . 6  |-  ( `' ( G  o.  F
) " x )  =  ( `' F " ( `' G "
x ) )
1615eleq1i 2346 . . . . 5  |-  ( ( `' ( G  o.  F ) " x
)  e.  J  <->  ( `' F " ( `' G " x ) )  e.  J )
1711, 16syl6bbr 254 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  (
( `' G "
x )  e.  ( J qTop  F )  <->  ( `' ( G  o.  F
) " x )  e.  J ) )
1817ralbidva 2559 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( A. x  e.  K  ( `' G " x )  e.  ( J qTop  F )  <->  A. x  e.  K  ( `' ( G  o.  F
) " x )  e.  J ) )
19 simprr 733 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  ->  G : Y --> Z )
2019biantrurd 494 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( A. x  e.  K  ( `' G " x )  e.  ( J qTop  F )  <->  ( G : Y --> Z  /\  A. x  e.  K  ( `' G " x )  e.  ( J qTop  F
) ) ) )
21 fof 5451 . . . . . 6  |-  ( F : X -onto-> Y  ->  F : X --> Y )
2221ad2antrl 708 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  ->  F : X --> Y )
23 fco 5398 . . . . 5  |-  ( ( G : Y --> Z  /\  F : X --> Y )  ->  ( G  o.  F ) : X --> Z )
2419, 22, 23syl2anc 642 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( G  o.  F
) : X --> Z )
2524biantrurd 494 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( A. x  e.  K  ( `' ( G  o.  F )
" x )  e.  J  <->  ( ( G  o.  F ) : X --> Z  /\  A. x  e.  K  ( `' ( G  o.  F ) " x
)  e.  J ) ) )
2618, 20, 253bitr3d 274 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( ( G : Y
--> Z  /\  A. x  e.  K  ( `' G " x )  e.  ( J qTop  F ) )  <->  ( ( G  o.  F ) : X --> Z  /\  A. x  e.  K  ( `' ( G  o.  F ) " x
)  e.  J ) ) )
27 qtoptopon 17395 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y )  ->  ( J qTop  F )  e.  (TopOn `  Y ) )
2827ad2ant2r 727 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( J qTop  F )  e.  (TopOn `  Y )
)
29 simplr 731 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  ->  K  e.  (TopOn `  Z
) )
30 iscn 16965 . . 3  |-  ( ( ( J qTop  F )  e.  (TopOn `  Y
)  /\  K  e.  (TopOn `  Z ) )  ->  ( G  e.  ( ( J qTop  F
)  Cn  K )  <-> 
( G : Y --> Z  /\  A. x  e.  K  ( `' G " x )  e.  ( J qTop  F ) ) ) )
3128, 29, 30syl2anc 642 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( G  e.  ( ( J qTop  F )  Cn  K )  <->  ( G : Y --> Z  /\  A. x  e.  K  ( `' G " x )  e.  ( J qTop  F
) ) ) )
32 iscn 16965 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  ->  ( ( G  o.  F )  e.  ( J  Cn  K
)  <->  ( ( G  o.  F ) : X --> Z  /\  A. x  e.  K  ( `' ( G  o.  F ) " x
)  e.  J ) ) )
3332adantr 451 . 2  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( ( G  o.  F )  e.  ( J  Cn  K )  <-> 
( ( G  o.  F ) : X --> Z  /\  A. x  e.  K  ( `' ( G  o.  F )
" x )  e.  J ) ) )
3426, 31, 333bitr4d 276 1  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( G  e.  ( ( J qTop  F )  Cn  K )  <->  ( G  o.  F )  e.  ( J  Cn  K ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   `'ccnv 4688   dom cdm 4689   "cima 4692    o. ccom 4693   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   qTop cqtop 13406  TopOnctopon 16632    Cn ccn 16954
This theorem is referenced by:  qtopeu  17407
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-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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-map 6774  df-qtop 13410  df-top 16636  df-topon 16639  df-cn 16957
  Copyright terms: Public domain W3C validator