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Theorem qtopval2 20024
Description: Value of the quotient topology function when  F is a function on the base set. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtopval.1  |-  X  = 
U. J
Assertion
Ref Expression
qtopval2  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( J qTop  F )  =  { s  e.  ~P Y  |  ( `' F " s )  e.  J } )
Distinct variable groups:    F, s    J, s    V, s    Y, s    Z, s    X, s

Proof of Theorem qtopval2
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  J  e.  V )
2 fof 5795 . . . . 5  |-  ( F : Z -onto-> Y  ->  F : Z --> Y )
323ad2ant2 1018 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z --> Y )
4 qtopval.1 . . . . . 6  |-  X  = 
U. J
5 uniexg 6582 . . . . . . 7  |-  ( J  e.  V  ->  U. J  e.  _V )
653ad2ant1 1017 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  U. J  e.  _V )
74, 6syl5eqel 2559 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
8 simp3 998 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
97, 8ssexd 4594 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
10 fex 6134 . . . 4  |-  ( ( F : Z --> Y  /\  Z  e.  _V )  ->  F  e.  _V )
113, 9, 10syl2anc 661 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F  e.  _V )
124qtopval 20023 . . 3  |-  ( ( J  e.  V  /\  F  e.  _V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
131, 11, 12syl2anc 661 . 2  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( J qTop  F )  =  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X )  e.  J } )
14 imassrn 5348 . . . . . 6  |-  ( F
" X )  C_  ran  F
15 forn 5798 . . . . . . 7  |-  ( F : Z -onto-> Y  ->  ran  F  =  Y )
16153ad2ant2 1018 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ran  F  =  Y )
1714, 16syl5sseq 3552 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  C_  Y )
18 foima 5800 . . . . . . 7  |-  ( F : Z -onto-> Y  -> 
( F " Z
)  =  Y )
19183ad2ant2 1018 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " Z
)  =  Y )
20 imass2 5372 . . . . . . 7  |-  ( Z 
C_  X  ->  ( F " Z )  C_  ( F " X ) )
218, 20syl 16 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " Z
)  C_  ( F " X ) )
2219, 21eqsstr3d 3539 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  C_  ( F " X ) )
2317, 22eqssd 3521 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  =  Y )
2423pweqd 4015 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ~P ( F " X
)  =  ~P Y
)
25 cnvimass 5357 . . . . . . 7  |-  ( `' F " s ) 
C_  dom  F
26 fdm 5735 . . . . . . . 8  |-  ( F : Z --> Y  ->  dom  F  =  Z )
273, 26syl 16 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  dom  F  =  Z )
2825, 27syl5sseq 3552 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  Z
)
2928, 8sstrd 3514 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  X
)
30 df-ss 3490 . . . . 5  |-  ( ( `' F " s ) 
C_  X  <->  ( ( `' F " s )  i^i  X )  =  ( `' F "
s ) )
3129, 30sylib 196 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( `' F " s )  i^i  X
)  =  ( `' F " s ) )
3231eleq1d 2536 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( ( `' F " s )  i^i  X )  e.  J  <->  ( `' F " s )  e.  J
) )
3324, 32rabeqbidv 3108 . 2  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  { s  e.  ~P ( F " X )  |  ( ( `' F " s )  i^i  X )  e.  J }  =  {
s  e.  ~P Y  |  ( `' F " s )  e.  J } )
3413, 33eqtrd 2508 1  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( J qTop  F )  =  { s  e.  ~P Y  |  ( `' F " s )  e.  J } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 973    = wceq 1379    e. wcel 1767   {crab 2818   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010   U.cuni 4245   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   -->wf 5584   -onto->wfo 5586  (class class class)co 6285   qTop cqtop 14761
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 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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 6288  df-oprab 6289  df-mpt2 6290  df-qtop 14765
This theorem is referenced by:  elqtop  20025
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