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Theorem qtopval2 17681
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 5612 . . . . 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 4665 . . . . . . 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 2488 . . . . 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 4310 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
10 fex 5928 . . . 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 17680 . . 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 5175 . . . . . 6  |-  ( F
" X )  C_  ran  F
15 forn 5615 . . . . . . 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 3356 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  C_  Y )
18 foima 5617 . . . . . . 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 5199 . . . . . . 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 3343 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  C_  ( F " X ) )
2317, 22eqssd 3325 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  =  Y )
2423pweqd 3764 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ~P ( F " X
)  =  ~P Y
)
25 cnvimass 5183 . . . . . . 7  |-  ( `' F " s ) 
C_  dom  F
26 fdm 5554 . . . . . . . 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 3356 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  Z
)
2928, 8sstrd 3318 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  X
)
30 df-ss 3294 . . . . 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 2470 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( ( `' F " s )  i^i  X )  e.  J  <->  ( `' F " s )  e.  J
) )
3324, 32rabeqbidv 2911 . 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 2436 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 1649    e. wcel 1721   {crab 2670   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   U.cuni 3975   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   -->wf 5409   -onto->wfo 5411  (class class class)co 6040   qTop cqtop 13684
This theorem is referenced by:  elqtop  17682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-qtop 13688
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