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Theorem dfac5lem3 7768
Description: Lemma for dfac5 7771. (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 4232 . . . 4  |-  { w }  e.  _V
2 vex 2804 . . . 4  |-  w  e. 
_V
31, 2xpex 4817 . . 3  |-  ( { w }  X.  w
)  e.  _V
4 neeq1 2467 . . . 4  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =/=  (/)  <->  ( {
w }  X.  w
)  =/=  (/) ) )
5 eqeq1 2302 . . . . 5  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =  ( { t }  X.  t )  <->  ( {
w }  X.  w
)  =  ( { t }  X.  t
) ) )
65rexbidv 2577 . . . 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 2927 . 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 2360 . 2  |-  ( ( { w }  X.  w )  e.  A  <->  ( { w }  X.  w )  e.  {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
11 xpeq2 4720 . . . . . 6  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  ( { w }  X.  (/) ) )
12 xp0 5114 . . . . . 6  |-  ( { w }  X.  (/) )  =  (/)
1311, 12syl6eq 2344 . . . . 5  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  (/) )
14 rneq 4920 . . . . . 6  |-  ( ( { w }  X.  w )  =  (/)  ->  ran  ( { w }  X.  w )  =  ran  (/) )
152snnz 3757 . . . . . . 7  |-  { w }  =/=  (/)
16 rnxp 5122 . . . . . . 7  |-  ( { w }  =/=  (/)  ->  ran  ( { w }  X.  w )  =  w )
1715, 16ax-mp 8 . . . . . 6  |-  ran  ( { w }  X.  w )  =  w
18 rn0 4952 . . . . . 6  |-  ran  (/)  =  (/)
1914, 17, 183eqtr3g 2351 . . . . 5  |-  ( ( { w }  X.  w )  =  (/)  ->  w  =  (/) )
2013, 19impbii 180 . . . 4  |-  ( w  =  (/)  <->  ( { w }  X.  w )  =  (/) )
2120necon3bii 2491 . . 3  |-  ( w  =/=  (/)  <->  ( { w }  X.  w )  =/=  (/) )
22 df-rex 2562 . . . 4  |-  ( E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t )  <->  E. t
( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
23 rneq 4920 . . . . . . . . . 10  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  ran  ( { w }  X.  w )  =  ran  ( { t }  X.  t ) )
24 vex 2804 . . . . . . . . . . . 12  |-  t  e. 
_V
2524snnz 3757 . . . . . . . . . . 11  |-  { t }  =/=  (/)
26 rnxp 5122 . . . . . . . . . . 11  |-  ( { t }  =/=  (/)  ->  ran  ( { t }  X.  t )  =  t )
2725, 26ax-mp 8 . . . . . . . . . 10  |-  ran  ( { t }  X.  t )  =  t
2823, 17, 273eqtr3g 2351 . . . . . . . . 9  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  w  =  t )
29 sneq 3664 . . . . . . . . . . 11  |-  ( w  =  t  ->  { w }  =  { t } )
3029xpeq1d 4728 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  w ) )
31 xpeq2 4720 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { t }  X.  w )  =  ( { t }  X.  t ) )
3230, 31eqtrd 2328 . . . . . . . . 9  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  t ) )
3328, 32impbii 180 . . . . . . . 8  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  <->  w  =  t )
34 equcom 1665 . . . . . . . 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 1572 . . . 4  |-  ( E. t ( t  e.  h  /\  ( { w }  X.  w
)  =  ( { t }  X.  t
) )  <->  E. t
( t  =  w  /\  t  e.  h
) )
40 elequ1 1699 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
412, 40ceqsexv 2836 . . . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   E.wrex 2557   (/)c0 3468   {csn 3653    X. cxp 4703   ran crn 4706
This theorem is referenced by:  dfac5lem5  7770
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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