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Theorem elqtop 17388
Description: Value of the quotient topology function. (Contributed by Mario Carneiro, 23-Mar-2015.)
Hypothesis
Ref Expression
qtopval.1  |-  X  = 
U. J
Assertion
Ref Expression
elqtop  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )

Proof of Theorem elqtop
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 qtopval.1 . . . 4  |-  X  = 
U. J
21qtopval2 17387 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( J qTop  F )  =  { s  e.  ~P Y  |  ( `' F " s )  e.  J } )
32eleq2d 2350 . 2  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( A  e.  ( J qTop  F )  <->  A  e.  { s  e.  ~P Y  |  ( `' F " s )  e.  J } ) )
4 imaeq2 5008 . . . . 5  |-  ( s  =  A  ->  ( `' F " s )  =  ( `' F " A ) )
54eleq1d 2349 . . . 4  |-  ( s  =  A  ->  (
( `' F "
s )  e.  J  <->  ( `' F " A )  e.  J ) )
65elrab 2923 . . 3  |-  ( A  e.  { s  e. 
~P Y  |  ( `' F " s )  e.  J }  <->  ( A  e.  ~P Y  /\  ( `' F " A )  e.  J ) )
7 simp3 957 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
8 uniexg 4517 . . . . . . . . 9  |-  ( J  e.  V  ->  U. J  e.  _V )
91, 8syl5eqel 2367 . . . . . . . 8  |-  ( J  e.  V  ->  X  e.  _V )
1093ad2ant1 976 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
11 ssexg 4160 . . . . . . 7  |-  ( ( Z  C_  X  /\  X  e.  _V )  ->  Z  e.  _V )
127, 10, 11syl2anc 642 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
13 simp2 956 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z -onto-> Y )
14 fornex 5750 . . . . . 6  |-  ( Z  e.  _V  ->  ( F : Z -onto-> Y  ->  Y  e.  _V )
)
1512, 13, 14sylc 56 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  e.  _V )
16 elpw2g 4174 . . . . 5  |-  ( Y  e.  _V  ->  ( A  e.  ~P Y  <->  A 
C_  Y ) )
1715, 16syl 15 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( A  e.  ~P Y 
<->  A  C_  Y )
)
1817anbi1d 685 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( ( A  e. 
~P Y  /\  ( `' F " A )  e.  J )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
196, 18syl5bb 248 . 2  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( A  e.  {
s  e.  ~P Y  |  ( `' F " s )  e.  J } 
<->  ( A  C_  Y  /\  ( `' F " A )  e.  J
) ) )
203, 19bitrd 244 1  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( A  e.  ( J qTop  F )  <->  ( A  C_  Y  /\  ( `' F " A )  e.  J ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   `'ccnv 4688   "cima 4692   -onto->wfo 5253  (class class class)co 5858   qTop cqtop 13406
This theorem is referenced by:  qtoptop2  17390  elqtop2  17392  elqtop3  17394
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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