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Theorem elqtop 19395
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 19394 . . 3  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  -> 
( J qTop  F )  =  { s  e.  ~P Y  |  ( `' F " s )  e.  J } )
32eleq2d 2521 . 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 5266 . . . . 5  |-  ( s  =  A  ->  ( `' F " s )  =  ( `' F " A ) )
54eleq1d 2520 . . . 4  |-  ( s  =  A  ->  (
( `' F "
s )  e.  J  <->  ( `' F " A )  e.  J ) )
65elrab 3217 . . 3  |-  ( A  e.  { s  e. 
~P Y  |  ( `' F " s )  e.  J }  <->  ( A  e.  ~P Y  /\  ( `' F " A )  e.  J ) )
7 uniexg 6480 . . . . . . . . 9  |-  ( J  e.  V  ->  U. J  e.  _V )
81, 7syl5eqel 2543 . . . . . . . 8  |-  ( J  e.  V  ->  X  e.  _V )
983ad2ant1 1009 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  X  e.  _V )
10 simp3 990 . . . . . . 7  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  C_  X )
119, 10ssexd 4540 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  Z  e.  _V )
12 simp2 989 . . . . . 6  |-  ( ( J  e.  V  /\  F : Z -onto-> Y  /\  Z  C_  X )  ->  F : Z -onto-> Y )
13 fornex 6649 . . . . . 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 4556 . . . . 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 704 . . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2799   _Vcvv 3071    C_ wss 3429   ~Pcpw 3961   U.cuni 4192   `'ccnv 4940   "cima 4944   -onto->wfo 5517  (class class class)co 6193   qTop cqtop 14552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-qtop 14556
This theorem is referenced by:  qtoptop2  19397  elqtop2  19399  elqtop3  19401
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