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Theorem dfac5lem2 7998
Description: Lemma for dfac5 8002. (Contributed by NM, 12-Apr-2004.)
Hypothesis
Ref Expression
dfac5lem.1  |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
Assertion
Ref Expression
dfac5lem2  |-  ( <.
w ,  g >.  e.  U. A  <->  ( w  e.  h  /\  g  e.  w ) )
Distinct variable groups:    w, u, t, h, g    w, A, g
Allowed substitution hints:    A( u, t, h)

Proof of Theorem dfac5lem2
StepHypRef Expression
1 dfac5lem.1 . . . 4  |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
21unieqi 4018 . . 3  |-  U. A  =  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
32eleq2i 2500 . 2  |-  ( <.
w ,  g >.  e.  U. A  <->  <. w ,  g >.  e.  U. {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
4 eluniab 4020 . . 3  |-  ( <.
w ,  g >.  e.  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }  <->  E. u
( <. w ,  g
>.  e.  u  /\  (
u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) ) )
5 r19.42v 2855 . . . . 5  |-  ( E. t  e.  h  ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) )
6 anass 631 . . . . 5  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  E. t  e.  h  u  =  ( { t }  X.  t ) )  <->  ( <. w ,  g >.  e.  u  /\  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) ) )
75, 6bitr2i 242 . . . 4  |-  ( (
<. w ,  g >.  e.  u  /\  (
u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) )  <->  E. t  e.  h  ( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )
87exbii 1592 . . 3  |-  ( E. u ( <. w ,  g >.  e.  u  /\  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) )  <->  E. u E. t  e.  h  ( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( {
t }  X.  t
) ) )
9 rexcom4 2968 . . . 4  |-  ( E. t  e.  h  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. u E. t  e.  h  ( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )
10 df-rex 2704 . . . 4  |-  ( E. t  e.  h  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. t ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) ) )
119, 10bitr3i 243 . . 3  |-  ( E. u E. t  e.  h  ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. t ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) ) )
124, 8, 113bitri 263 . 2  |-  ( <.
w ,  g >.  e.  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }  <->  E. t
( t  e.  h  /\  E. u ( (
<. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) ) )
13 ancom 438 . . . . . . . . 9  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
14 ne0i 3627 . . . . . . . . . . 11  |-  ( <.
w ,  g >.  e.  u  ->  u  =/=  (/) )
1514pm4.71i 614 . . . . . . . . . 10  |-  ( <.
w ,  g >.  e.  u  <->  ( <. w ,  g >.  e.  u  /\  u  =/=  (/) ) )
1615anbi2i 676 . . . . . . . . 9  |-  ( ( u  =  ( { t }  X.  t
)  /\  <. w ,  g >.  e.  u
)  <->  ( u  =  ( { t }  X.  t )  /\  ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) ) ) )
1713, 16bitr4i 244 . . . . . . . 8  |-  ( ( ( <. w ,  g
>.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u ) )
1817exbii 1592 . . . . . . 7  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. u ( u  =  ( { t }  X.  t )  /\  <. w ,  g
>.  e.  u ) )
19 snex 4398 . . . . . . . . 9  |-  { t }  e.  _V
20 vex 2952 . . . . . . . . 9  |-  t  e. 
_V
2119, 20xpex 4983 . . . . . . . 8  |-  ( { t }  X.  t
)  e.  _V
22 eleq2 2497 . . . . . . . 8  |-  ( u  =  ( { t }  X.  t )  ->  ( <. w ,  g >.  e.  u  <->  <.
w ,  g >.  e.  ( { t }  X.  t ) ) )
2321, 22ceqsexv 2984 . . . . . . 7  |-  ( E. u ( u  =  ( { t }  X.  t )  /\  <.
w ,  g >.  e.  u )  <->  <. w ,  g >.  e.  ( { t }  X.  t ) )
2418, 23bitri 241 . . . . . 6  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  <. w ,  g
>.  e.  ( { t }  X.  t ) )
2524anbi2i 676 . . . . 5  |-  ( ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )  <->  ( t  e.  h  /\  <. w ,  g >.  e.  ( { t }  X.  t ) ) )
26 opelxp 4901 . . . . . . 7  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( w  e.  { t }  /\  g  e.  t )
)
27 elsn 3822 . . . . . . . . 9  |-  ( w  e.  { t }  <-> 
w  =  t )
28 equcom 1692 . . . . . . . . 9  |-  ( w  =  t  <->  t  =  w )
2927, 28bitri 241 . . . . . . . 8  |-  ( w  e.  { t }  <-> 
t  =  w )
3029anbi1i 677 . . . . . . 7  |-  ( ( w  e.  { t }  /\  g  e.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3126, 30bitri 241 . . . . . 6  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( t  =  w  /\  g  e.  t ) )
3231anbi2i 676 . . . . 5  |-  ( ( t  e.  h  /\  <.
w ,  g >.  e.  ( { t }  X.  t ) )  <-> 
( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) ) )
33 an12 773 . . . . 5  |-  ( ( t  e.  h  /\  ( t  =  w  /\  g  e.  t ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3425, 32, 333bitri 263 . . . 4  |-  ( ( t  e.  h  /\  E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) ) )  <->  ( t  =  w  /\  (
t  e.  h  /\  g  e.  t )
) )
3534exbii 1592 . . 3  |-  ( E. t ( t  e.  h  /\  E. u
( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( {
t }  X.  t
) ) )  <->  E. t
( t  =  w  /\  ( t  e.  h  /\  g  e.  t ) ) )
36 vex 2952 . . . 4  |-  w  e. 
_V
37 elequ1 1728 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
38 eleq2 2497 . . . . 5  |-  ( t  =  w  ->  (
g  e.  t  <->  g  e.  w ) )
3937, 38anbi12d 692 . . . 4  |-  ( t  =  w  ->  (
( t  e.  h  /\  g  e.  t
)  <->  ( w  e.  h  /\  g  e.  w ) ) )
4036, 39ceqsexv 2984 . . 3  |-  ( E. t ( t  =  w  /\  ( t  e.  h  /\  g  e.  t ) )  <->  ( w  e.  h  /\  g  e.  w ) )
4135, 40bitri 241 . 2  |-  ( E. t ( t  e.  h  /\  E. u
( ( <. w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( {
t }  X.  t
) ) )  <->  ( w  e.  h  /\  g  e.  w ) )
423, 12, 413bitri 263 1  |-  ( <.
w ,  g >.  e.  U. A  <->  ( w  e.  h  /\  g  e.  w ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   E.wrex 2699   (/)c0 3621   {csn 3807   <.cop 3810   U.cuni 4008    X. cxp 4869
This theorem is referenced by:  dfac5lem5  8001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-opab 4260  df-xp 4877
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