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Theorem elqtop 20364
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 20363 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( J qTop  F )  =  { s  e.  ~P Y  |  ( `' F " s )  e.  J } )
32eleq2d 2524 . 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 5321 . . . . 5  |-  ( s  =  A  ->  ( `' F " s )  =  ( `' F " A ) )
54eleq1d 2523 . . . 4  |-  ( s  =  A  ->  (
( `' F "
s )  e.  J  <->  ( `' F " A )  e.  J ) )
65elrab 3254 . . 3  |-  ( A  e.  { s  e. 
~P Y  |  ( `' F " s )  e.  J }  <->  ( A  e.  ~P Y  /\  ( `' F " A )  e.  J ) )
7 uniexg 6570 . . . . . . . . 9  |-  ( J  e.  V  ->  U. J  e.  _V )
81, 7syl5eqel 2546 . . . . . . . 8  |-  ( J  e.  V  ->  X  e.  _V )
983ad2ant1 1015 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
10 simp3 996 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
119, 10ssexd 4584 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
12 simp2 995 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z -onto-> Y )
13 fornex 6742 . . . . . 6  |-  ( Z  e.  _V  ->  ( F : Z -onto-> Y  ->  Y  e.  _V )
)
1411, 12, 13sylc 60 . . . . 5  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Y  e.  _V )
15 elpw2g 4600 . . . . 5  |-  ( Y  e.  _V  ->  ( A  e.  ~P Y  <->  A 
C_  Y ) )
1614, 15syl 16 . . . 4  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( A  e.  ~P Y 
<->  A  C_  Y )
)
1716anbi1d 702 . . 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 257 . 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 253 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 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    C_ wss 3461   ~Pcpw 3999   U.cuni 4235   `'ccnv 4987   "cima 4991   -onto->wfo 5568  (class class class)co 6270   qTop cqtop 14992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-qtop 14996
This theorem is referenced by:  qtoptop2  20366  elqtop2  20368  elqtop3  20370
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