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Theorem qtopval2 17387
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 955 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  J  e.  V )
2 fof 5451 . . . . 5  |-  ( F : Z -onto-> Y  ->  F : Z --> Y )
323ad2ant2 977 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z --> Y )
4 simp3 957 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
5 qtopval.1 . . . . . 6  |-  X  = 
U. J
6 uniexg 4517 . . . . . . 7  |-  ( J  e.  V  ->  U. J  e.  _V )
763ad2ant1 976 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  U. J  e.  _V )
85, 7syl5eqel 2367 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
9 ssexg 4160 . . . . 5  |-  ( ( Z  C_  X  /\  X  e.  _V )  ->  Z  e.  _V )
104, 8, 9syl2anc 642 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
11 fex 5749 . . . 4  |-  ( ( F : Z --> Y  /\  Z  e.  _V )  ->  F  e.  _V )
123, 10, 11syl2anc 642 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F  e.  _V )
135qtopval 17386 . . 3  |-  ( ( J  e.  V  /\  F  e.  _V )  ->  ( J qTop  F )  =  { s  e. 
~P ( F " X )  |  ( ( `' F "
s )  i^i  X
)  e.  J }
)
141, 12, 13syl2anc 642 . 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 } )
15 imassrn 5025 . . . . . 6  |-  ( F
" X )  C_  ran  F
16 forn 5454 . . . . . . 7  |-  ( F : Z -onto-> Y  ->  ran  F  =  Y )
17163ad2ant2 977 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ran  F  =  Y )
1815, 17syl5sseq 3226 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  C_  Y )
19 foima 5456 . . . . . . 7  |-  ( F : Z -onto-> Y  -> 
( F " Z
)  =  Y )
20193ad2ant2 977 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " Z
)  =  Y )
21 imass2 5049 . . . . . . 7  |-  ( Z 
C_  X  ->  ( F " Z )  C_  ( F " X ) )
224, 21syl 15 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " Z
)  C_  ( F " X ) )
2320, 22eqsstr3d 3213 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  C_  ( F " X ) )
2418, 23eqssd 3196 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( F " X
)  =  Y )
2524pweqd 3630 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  ~P ( F " X
)  =  ~P Y
)
26 cnvimass 5033 . . . . . . 7  |-  ( `' F " s ) 
C_  dom  F
27 fdm 5393 . . . . . . . 8  |-  ( F : Z --> Y  ->  dom  F  =  Z )
283, 27syl 15 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  dom  F  =  Z )
2926, 28syl5sseq 3226 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  Z
)
3029, 4sstrd 3189 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( `' F "
s )  C_  X
)
31 df-ss 3166 . . . . 5  |-  ( ( `' F " s ) 
C_  X  <->  ( ( `' F " s )  i^i  X )  =  ( `' F "
s ) )
3230, 31sylib 188 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( `' F " s )  i^i  X
)  =  ( `' F " s ) )
3332eleq1d 2349 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( ( `' F " s )  i^i  X )  e.  J  <->  ( `' F " s )  e.  J
) )
3425, 33rabeqbidv 2783 . 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 } )
3514, 34eqtrd 2315 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 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   -->wf 5251   -onto->wfo 5253  (class class class)co 5858   qTop cqtop 13406
This theorem is referenced by:  elqtop  17388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-qtop 13410
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