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Theorem dfac5lem2 7931
Description: Lemma for dfac5 7935. (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 3960 . . 3  |-  U. A  =  U. { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
32eleq2i 2444 . 2  |-  ( <.
w ,  g >.  e.  U. A  <->  <. w ,  g >.  e.  U. {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
4 eluniab 3962 . . 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 2798 . . . . 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 1589 . . 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 2911 . . . 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 2648 . . . 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 3570 . . . . . . . . . . 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 1589 . . . . . . 7  |-  ( E. u ( ( <.
w ,  g >.  e.  u  /\  u  =/=  (/) )  /\  u  =  ( { t }  X.  t ) )  <->  E. u ( u  =  ( { t }  X.  t )  /\  <. w ,  g
>.  e.  u ) )
19 snex 4339 . . . . . . . . 9  |-  { t }  e.  _V
20 vex 2895 . . . . . . . . 9  |-  t  e. 
_V
2119, 20xpex 4923 . . . . . . . 8  |-  ( { t }  X.  t
)  e.  _V
22 eleq2 2441 . . . . . . . 8  |-  ( u  =  ( { t }  X.  t )  ->  ( <. w ,  g >.  e.  u  <->  <.
w ,  g >.  e.  ( { t }  X.  t ) ) )
2321, 22ceqsexv 2927 . . . . . . 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 4841 . . . . . . 7  |-  ( <.
w ,  g >.  e.  ( { t }  X.  t )  <->  ( w  e.  { t }  /\  g  e.  t )
)
27 elsn 3765 . . . . . . . . 9  |-  ( w  e.  { t }  <-> 
w  =  t )
28 equcom 1687 . . . . . . . . 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 1589 . . 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 2895 . . . 4  |-  w  e. 
_V
37 elequ1 1720 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
38 eleq2 2441 . . . . 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 2927 . . 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 1547    = wceq 1649    e. wcel 1717   {cab 2366    =/= wne 2543   E.wrex 2643   (/)c0 3564   {csn 3750   <.cop 3753   U.cuni 3950    X. cxp 4809
This theorem is referenced by:  dfac5lem5  7934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-opab 4201  df-xp 4817
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