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Theorem qtopcn 19943
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 19932 . . . . . . 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 5348 . . . . . . . 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 5726 . . . . . . . . 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 3545 . . . . . . 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 5179 . . . . . . . 8  |-  `' ( G  o.  F )  =  ( `' F  o.  `' G )
1312imaeq1i 5325 . . . . . . 7  |-  ( `' ( G  o.  F
) " x )  =  ( ( `' F  o.  `' G
) " x )
14 imaco 5503 . . . . . . 7  |-  ( ( `' F  o.  `' G ) " x
)  =  ( `' F " ( `' G " x ) )
1513, 14eqtri 2489 . . . . . 6  |-  ( `' ( G  o.  F
) " x )  =  ( `' F " ( `' G "
x ) )
1615eleq1i 2537 . . . . 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 2893 . . 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 5786 . . . . . 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 5732 . . . . 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 19933 . . . 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 19495 . . 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 19495 . . 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 1374    e. wcel 1762   A.wral 2807    C_ wss 3469   `'ccnv 4991   dom cdm 4992   "cima 4995    o. ccom 4996   -->wf 5575   -onto->wfo 5577   ` cfv 5579  (class class class)co 6275   qTop cqtop 14747  TopOnctopon 19155    Cn ccn 19484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-qtop 14751  df-top 19159  df-topon 19162  df-cn 19487
This theorem is referenced by:  qtopeu  19945
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