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Theorem dfac5lem3 7752
Description: Lemma for dfac5 7755. (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 4216 . . . 4  |-  { w }  e.  _V
2 vex 2791 . . . 4  |-  w  e. 
_V
31, 2xpex 4801 . . 3  |-  ( { w }  X.  w
)  e.  _V
4 neeq1 2454 . . . 4  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =/=  (/)  <->  ( {
w }  X.  w
)  =/=  (/) ) )
5 eqeq1 2289 . . . . 5  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =  ( { t }  X.  t )  <->  ( {
w }  X.  w
)  =  ( { t }  X.  t
) ) )
65rexbidv 2564 . . . 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 691 . . 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 2914 . 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 2347 . 2  |-  ( ( { w }  X.  w )  e.  A  <->  ( { w }  X.  w )  e.  {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
11 xpeq2 4704 . . . . . 6  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  ( { w }  X.  (/) ) )
12 xp0 5098 . . . . . 6  |-  ( { w }  X.  (/) )  =  (/)
1311, 12syl6eq 2331 . . . . 5  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  (/) )
14 rneq 4904 . . . . . 6  |-  ( ( { w }  X.  w )  =  (/)  ->  ran  ( { w }  X.  w )  =  ran  (/) )
152snnz 3744 . . . . . . 7  |-  { w }  =/=  (/)
16 rnxp 5106 . . . . . . 7  |-  ( { w }  =/=  (/)  ->  ran  ( { w }  X.  w )  =  w )
1715, 16ax-mp 8 . . . . . 6  |-  ran  ( { w }  X.  w )  =  w
18 rn0 4936 . . . . . 6  |-  ran  (/)  =  (/)
1914, 17, 183eqtr3g 2338 . . . . 5  |-  ( ( { w }  X.  w )  =  (/)  ->  w  =  (/) )
2013, 19impbii 180 . . . 4  |-  ( w  =  (/)  <->  ( { w }  X.  w )  =  (/) )
2120necon3bii 2478 . . 3  |-  ( w  =/=  (/)  <->  ( { w }  X.  w )  =/=  (/) )
22 df-rex 2549 . . . 4  |-  ( E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t )  <->  E. t
( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
23 rneq 4904 . . . . . . . . . 10  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  ran  ( { w }  X.  w )  =  ran  ( { t }  X.  t ) )
24 vex 2791 . . . . . . . . . . . 12  |-  t  e. 
_V
2524snnz 3744 . . . . . . . . . . 11  |-  { t }  =/=  (/)
26 rnxp 5106 . . . . . . . . . . 11  |-  ( { t }  =/=  (/)  ->  ran  ( { t }  X.  t )  =  t )
2725, 26ax-mp 8 . . . . . . . . . 10  |-  ran  ( { t }  X.  t )  =  t
2823, 17, 273eqtr3g 2338 . . . . . . . . 9  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  w  =  t )
29 sneq 3651 . . . . . . . . . . 11  |-  ( w  =  t  ->  { w }  =  { t } )
3029xpeq1d 4712 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  w ) )
31 xpeq2 4704 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { t }  X.  w )  =  ( { t }  X.  t ) )
3230, 31eqtrd 2315 . . . . . . . . 9  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  t ) )
3328, 32impbii 180 . . . . . . . 8  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  <->  w  =  t )
34 equcom 1647 . . . . . . . 8  |-  ( w  =  t  <->  t  =  w )
3533, 34bitri 240 . . . . . . 7  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  <->  t  =  w )
3635anbi2i 675 . . . . . 6  |-  ( ( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) )  <->  ( t  e.  h  /\  t  =  w ) )
37 ancom 437 . . . . . 6  |-  ( ( t  e.  h  /\  t  =  w )  <->  ( t  =  w  /\  t  e.  h )
)
3836, 37bitri 240 . . . . 5  |-  ( ( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) )  <->  ( t  =  w  /\  t  e.  h ) )
3938exbii 1569 . . . 4  |-  ( E. t ( t  e.  h  /\  ( { w }  X.  w
)  =  ( { t }  X.  t
) )  <->  E. t
( t  =  w  /\  t  e.  h
) )
40 elequ1 1687 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
412, 40ceqsexv 2823 . . . 4  |-  ( E. t ( t  =  w  /\  t  e.  h )  <->  w  e.  h )
4222, 39, 413bitrri 263 . . 3  |-  ( w  e.  h  <->  E. t  e.  h  ( {
w }  X.  w
)  =  ( { t }  X.  t
) )
4321, 42anbi12i 678 . 2  |-  ( ( w  =/=  (/)  /\  w  e.  h )  <->  ( ( { w }  X.  w )  =/=  (/)  /\  E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
448, 10, 433bitr4i 268 1  |-  ( ( { w }  X.  w )  e.  A  <->  ( w  =/=  (/)  /\  w  e.  h ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   E.wrex 2544   (/)c0 3455   {csn 3640    X. cxp 4687   ran crn 4690
This theorem is referenced by:  dfac5lem5  7754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700
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