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Theorem qtopcn 19405
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 757 . . . . . . 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 759 . . . . . . 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 19394 . . . . . . 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 661 . . . . . 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 5289 . . . . . . . 8  |-  ( `' G " x ) 
C_  dom  G
6 simplrr 760 . . . . . . . . 9  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  G : Y --> Z )
7 fdm 5663 . . . . . . . . 9  |-  ( G : Y --> Z  ->  dom  G  =  Y )
86, 7syl 16 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  dom  G  =  Y )
95, 8syl5sseq 3504 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z ) )  /\  ( F : X -onto-> Y  /\  G : Y --> Z ) )  /\  x  e.  K )  ->  ( `' G " x ) 
C_  Y )
109biantrurd 508 . . . . . 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 256 . . . . 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 5125 . . . . . . . 8  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
1312imaeq1i 5266 . . . . . . 7  |-  ( `' ( G  o.  F
) " x )  =  ( ( `' F  o.  `' G
) " x )
14 imaco 5443 . . . . . . 7  |-  ( ( `' F  o.  `' G ) " x
)  =  ( `' F " ( `' G " x ) )
1513, 14eqtri 2480 . . . . . 6  |-  ( `' ( G  o.  F
) " x )  =  ( `' F " ( `' G "
x ) )
1615eleq1i 2528 . . . . 5  |-  ( ( `' ( G  o.  F ) " x
)  e.  J  <->  ( `' F " ( `' G " x ) )  e.  J )
1711, 16syl6bbr 263 . . . 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 2836 . . 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 756 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  ->  G : Y --> Z )
2019biantrurd 508 . . 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 5720 . . . . . 6  |-  ( F : X -onto-> Y  ->  F : X --> Y )
2221ad2antrl 727 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  ->  F : X --> Y )
23 fco 5668 . . . . 5  |-  ( ( G : Y --> Z  /\  F : X --> Y )  ->  ( G  o.  F ) : X --> Z )
2419, 22, 23syl2anc 661 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( G  o.  F
) : X --> Z )
2524biantrurd 508 . . 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 283 . 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 19395 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  F : X -onto-> Y )  ->  ( J qTop  F )  e.  (TopOn `  Y ) )
2827ad2ant2r 746 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  -> 
( J qTop  F )  e.  (TopOn `  Y )
)
29 simplr 754 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Z )
)  /\  ( F : X -onto-> Y  /\  G : Y
--> Z ) )  ->  K  e.  (TopOn `  Z
) )
30 iscn 18957 . . 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 661 . 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 18957 . . 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 465 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795    C_ wss 3428   `'ccnv 4939   dom cdm 4940   "cima 4943    o. ccom 4944   -->wf 5514   -onto->wfo 5516   ` cfv 5518  (class class class)co 6192   qTop cqtop 14545  TopOnctopon 18617    Cn ccn 18946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-map 7318  df-qtop 14549  df-top 18621  df-topon 18624  df-cn 18949
This theorem is referenced by:  qtopeu  19407
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