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Theorem dfac5lem3 7940
Description: Lemma for dfac5 7943. (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
dfac5lem3  |-  ( ( { w }  X.  w )  e.  A  <->  ( w  =/=  (/)  /\  w  e.  h ) )
Distinct variable groups:    w, u, t, h    w, A
Allowed substitution hints:    A( u, t, h)

Proof of Theorem dfac5lem3
StepHypRef Expression
1 snex 4347 . . . 4  |-  { w }  e.  _V
2 vex 2903 . . . 4  |-  w  e. 
_V
31, 2xpex 4931 . . 3  |-  ( { w }  X.  w
)  e.  _V
4 neeq1 2559 . . . 4  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =/=  (/)  <->  ( {
w }  X.  w
)  =/=  (/) ) )
5 eqeq1 2394 . . . . 5  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =  ( { t }  X.  t )  <->  ( {
w }  X.  w
)  =  ( { t }  X.  t
) ) )
65rexbidv 2671 . . . 4  |-  ( u  =  ( { w }  X.  w )  -> 
( E. t  e.  h  u  =  ( { t }  X.  t )  <->  E. t  e.  h  ( {
w }  X.  w
)  =  ( { t }  X.  t
) ) )
74, 6anbi12d 692 . . 3  |-  ( u  =  ( { w }  X.  w )  -> 
( ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) )  <->  ( ( { w }  X.  w )  =/=  (/)  /\  E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t ) ) ) )
83, 7elab 3026 . 2  |-  ( ( { w }  X.  w )  e.  {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }  <-> 
( ( { w }  X.  w )  =/=  (/)  /\  E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
9 dfac5lem.1 . . 3  |-  A  =  { u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) }
109eleq2i 2452 . 2  |-  ( ( { w }  X.  w )  e.  A  <->  ( { w }  X.  w )  e.  {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
11 xpeq2 4834 . . . . . 6  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  ( { w }  X.  (/) ) )
12 xp0 5232 . . . . . 6  |-  ( { w }  X.  (/) )  =  (/)
1311, 12syl6eq 2436 . . . . 5  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  (/) )
14 rneq 5036 . . . . . 6  |-  ( ( { w }  X.  w )  =  (/)  ->  ran  ( { w }  X.  w )  =  ran  (/) )
152snnz 3866 . . . . . . 7  |-  { w }  =/=  (/)
16 rnxp 5240 . . . . . . 7  |-  ( { w }  =/=  (/)  ->  ran  ( { w }  X.  w )  =  w )
1715, 16ax-mp 8 . . . . . 6  |-  ran  ( { w }  X.  w )  =  w
18 rn0 5068 . . . . . 6  |-  ran  (/)  =  (/)
1914, 17, 183eqtr3g 2443 . . . . 5  |-  ( ( { w }  X.  w )  =  (/)  ->  w  =  (/) )
2013, 19impbii 181 . . . 4  |-  ( w  =  (/)  <->  ( { w }  X.  w )  =  (/) )
2120necon3bii 2583 . . 3  |-  ( w  =/=  (/)  <->  ( { w }  X.  w )  =/=  (/) )
22 df-rex 2656 . . . 4  |-  ( E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t )  <->  E. t
( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
23 rneq 5036 . . . . . . . . . 10  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  ran  ( { w }  X.  w )  =  ran  ( { t }  X.  t ) )
24 vex 2903 . . . . . . . . . . . 12  |-  t  e. 
_V
2524snnz 3866 . . . . . . . . . . 11  |-  { t }  =/=  (/)
26 rnxp 5240 . . . . . . . . . . 11  |-  ( { t }  =/=  (/)  ->  ran  ( { t }  X.  t )  =  t )
2725, 26ax-mp 8 . . . . . . . . . 10  |-  ran  ( { t }  X.  t )  =  t
2823, 17, 273eqtr3g 2443 . . . . . . . . 9  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  w  =  t )
29 sneq 3769 . . . . . . . . . . 11  |-  ( w  =  t  ->  { w }  =  { t } )
3029xpeq1d 4842 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  w ) )
31 xpeq2 4834 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { t }  X.  w )  =  ( { t }  X.  t ) )
3230, 31eqtrd 2420 . . . . . . . . 9  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  t ) )
3328, 32impbii 181 . . . . . . . 8  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  <->  w  =  t )
34 equcom 1687 . . . . . . . 8  |-  ( w  =  t  <->  t  =  w )
3533, 34bitri 241 . . . . . . 7  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  <->  t  =  w )
3635anbi2i 676 . . . . . 6  |-  ( ( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) )  <->  ( t  e.  h  /\  t  =  w ) )
37 ancom 438 . . . . . 6  |-  ( ( t  e.  h  /\  t  =  w )  <->  ( t  =  w  /\  t  e.  h )
)
3836, 37bitri 241 . . . . 5  |-  ( ( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) )  <->  ( t  =  w  /\  t  e.  h ) )
3938exbii 1589 . . . 4  |-  ( E. t ( t  e.  h  /\  ( { w }  X.  w
)  =  ( { t }  X.  t
) )  <->  E. t
( t  =  w  /\  t  e.  h
) )
40 elequ1 1720 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
412, 40ceqsexv 2935 . . . 4  |-  ( E. t ( t  =  w  /\  t  e.  h )  <->  w  e.  h )
4222, 39, 413bitrri 264 . . 3  |-  ( w  e.  h  <->  E. t  e.  h  ( {
w }  X.  w
)  =  ( { t }  X.  t
) )
4321, 42anbi12i 679 . 2  |-  ( ( w  =/=  (/)  /\  w  e.  h )  <->  ( ( { w }  X.  w )  =/=  (/)  /\  E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
448, 10, 433bitr4i 269 1  |-  ( ( { w }  X.  w )  e.  A  <->  ( w  =/=  (/)  /\  w  e.  h ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374    =/= wne 2551   E.wrex 2651   (/)c0 3572   {csn 3758    X. cxp 4817   ran crn 4820
This theorem is referenced by:  dfac5lem5  7942
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 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827  df-dm 4829  df-rn 4830
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