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Theorem qtopval2 17728
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 957 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  J  e.  V )
2 fof 5653 . . . . 5  |-  ( F : Z -onto-> Y  ->  F : Z --> Y )
323ad2ant2 979 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z --> Y )
4 qtopval.1 . . . . . 6  |-  X  = 
U. J
5 uniexg 4706 . . . . . . 7  |-  ( J  e.  V  ->  U. J  e.  _V )
653ad2ant1 978 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  U. J  e.  _V )
74, 6syl5eqel 2520 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
8 simp3 959 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
97, 8ssexd 4350 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
10 fex 5969 . . . 4  |-  ( ( F : Z --> Y  /\  Z  e.  _V )  ->  F  e.  _V )
113, 9, 10syl2anc 643 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F  e.  _V )
124qtopval 17727 . . 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 643 . 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 5216 . . . . . 6  |-  ( F
" X )  C_  ran  F
15 forn 5656 . . . . . . 7  |-  ( F : Z -onto-> Y  ->  ran  F  =  Y )
16153ad2ant2 979 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ran  F  =  Y )
1714, 16syl5sseq 3396 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  C_  Y )
18 foima 5658 . . . . . . 7  |-  ( F : Z -onto-> Y  -> 
( F " Z
)  =  Y )
19183ad2ant2 979 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " Z
)  =  Y )
20 imass2 5240 . . . . . . 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 3383 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  C_  ( F " X ) )
2317, 22eqssd 3365 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  =  Y )
2423pweqd 3804 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ~P ( F " X
)  =  ~P Y
)
25 cnvimass 5224 . . . . . . 7  |-  ( `' F " s ) 
C_  dom  F
26 fdm 5595 . . . . . . . 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 3396 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  Z
)
2928, 8sstrd 3358 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  X
)
30 df-ss 3334 . . . . 5  |-  ( ( `' F " s ) 
C_  X  <->  ( ( `' F " s )  i^i  X )  =  ( `' F "
s ) )
3129, 30sylib 189 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( `' F " s )  i^i  X
)  =  ( `' F " s ) )
3231eleq1d 2502 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( ( `' F " s )  i^i  X )  e.  J  <->  ( `' F " s )  e.  J
) )
3324, 32rabeqbidv 2951 . 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 2468 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 set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    i^i cin 3319    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881   -->wf 5450   -onto->wfo 5452  (class class class)co 6081   qTop cqtop 13729
This theorem is referenced by:  elqtop  17729
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-qtop 13733
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