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Theorem elqtop 20761
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 20760 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( J qTop  F )  =  { s  e.  ~P Y  |  ( `' F " s )  e.  J } )
32eleq2d 2525 . 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 5183 . . . . 5  |-  ( s  =  A  ->  ( `' F " s )  =  ( `' F " A ) )
54eleq1d 2524 . . . 4  |-  ( s  =  A  ->  (
( `' F "
s )  e.  J  <->  ( `' F " A )  e.  J ) )
65elrab 3208 . . 3  |-  ( A  e.  { s  e. 
~P Y  |  ( `' F " s )  e.  J }  <->  ( A  e.  ~P Y  /\  ( `' F " A )  e.  J ) )
7 uniexg 6615 . . . . . . . . 9  |-  ( J  e.  V  ->  U. J  e.  _V )
81, 7syl5eqel 2544 . . . . . . . 8  |-  ( J  e.  V  ->  X  e.  _V )
983ad2ant1 1035 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
10 simp3 1016 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
119, 10ssexd 4564 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
12 simp2 1015 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z -onto-> Y )
13 fornex 6789 . . . . . 6  |-  ( Z  e.  _V  ->  ( F : Z -onto-> Y  ->  Y  e.  _V )
)
1411, 12, 13sylc 62 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  e.  _V )
15 elpw2g 4580 . . . . 5  |-  ( Y  e.  _V  ->  ( A  e.  ~P Y  <->  A 
C_  Y ) )
1614, 15syl 17 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( A  e.  ~P Y 
<->  A  C_  Y )
)
1716anbi1d 716 . . 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 ) ) )
186, 17syl5bb 265 . 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
) ) )
193, 18bitrd 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    = wceq 1455    e. wcel 1898   {crab 2753   _Vcvv 3057    C_ wss 3416   ~Pcpw 3963   U.cuni 4212   `'ccnv 4852   "cima 4856   -onto->wfo 5599  (class class class)co 6315   qTop cqtop 15450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-qtop 15455
This theorem is referenced by:  qtoptop2  20763  elqtop2  20765  elqtop3  20767
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