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Theorem qtopval2 20363
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 994 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  J  e.  V )
2 fof 5777 . . . . 5  |-  ( F : Z -onto-> Y  ->  F : Z --> Y )
323ad2ant2 1016 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z --> Y )
4 qtopval.1 . . . . . 6  |-  X  = 
U. J
5 uniexg 6570 . . . . . . 7  |-  ( J  e.  V  ->  U. J  e.  _V )
653ad2ant1 1015 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  U. J  e.  _V )
74, 6syl5eqel 2546 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
8 simp3 996 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
97, 8ssexd 4584 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
10 fex 6120 . . . 4  |-  ( ( F : Z --> Y  /\  Z  e.  _V )  ->  F  e.  _V )
113, 9, 10syl2anc 659 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F  e.  _V )
124qtopval 20362 . . 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 659 . 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 5336 . . . . . 6  |-  ( F
" X )  C_  ran  F
15 forn 5780 . . . . . . 7  |-  ( F : Z -onto-> Y  ->  ran  F  =  Y )
16153ad2ant2 1016 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ran  F  =  Y )
1714, 16syl5sseq 3537 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  C_  Y )
18 foima 5782 . . . . . . 7  |-  ( F : Z -onto-> Y  -> 
( F " Z
)  =  Y )
19183ad2ant2 1016 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " Z
)  =  Y )
20 imass2 5360 . . . . . . 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 3524 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  C_  ( F " X ) )
2317, 22eqssd 3506 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  =  Y )
2423pweqd 4004 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ~P ( F " X
)  =  ~P Y
)
25 cnvimass 5345 . . . . . . 7  |-  ( `' F " s ) 
C_  dom  F
26 fdm 5717 . . . . . . . 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 3537 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  Z
)
2928, 8sstrd 3499 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  X
)
30 df-ss 3475 . . . . 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 2523 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( ( `' F " s )  i^i  X )  e.  J  <->  ( `' F " s )  e.  J
) )
3324, 32rabeqbidv 3101 . 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 2495 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 971    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    i^i cin 3460    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   `'ccnv 4987   dom cdm 4988   ran crn 4989   "cima 4991   -->wf 5566   -onto->wfo 5568  (class class class)co 6270   qTop cqtop 14992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-qtop 14996
This theorem is referenced by:  elqtop  20364
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