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Theorem dfac5lem3 8581
Description: Lemma for dfac5 8584. (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 4654 . . . 4  |-  { w }  e.  _V
2 vex 3059 . . . 4  |-  w  e. 
_V
31, 2xpex 6621 . . 3  |-  ( { w }  X.  w
)  e.  _V
4 neeq1 2697 . . . 4  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =/=  (/)  <->  ( {
w }  X.  w
)  =/=  (/) ) )
5 eqeq1 2465 . . . . 5  |-  ( u  =  ( { w }  X.  w )  -> 
( u  =  ( { t }  X.  t )  <->  ( {
w }  X.  w
)  =  ( { t }  X.  t
) ) )
65rexbidv 2912 . . . 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 722 . . 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 3196 . 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 2531 . 2  |-  ( ( { w }  X.  w )  e.  A  <->  ( { w }  X.  w )  e.  {
u  |  ( u  =/=  (/)  /\  E. t  e.  h  u  =  ( { t }  X.  t ) ) } )
11 xpeq2 4867 . . . . . 6  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  ( { w }  X.  (/) ) )
12 xp0 5273 . . . . . 6  |-  ( { w }  X.  (/) )  =  (/)
1311, 12syl6eq 2511 . . . . 5  |-  ( w  =  (/)  ->  ( { w }  X.  w
)  =  (/) )
14 rneq 5078 . . . . . 6  |-  ( ( { w }  X.  w )  =  (/)  ->  ran  ( { w }  X.  w )  =  ran  (/) )
152snnz 4102 . . . . . . 7  |-  { w }  =/=  (/)
16 rnxp 5285 . . . . . . 7  |-  ( { w }  =/=  (/)  ->  ran  ( { w }  X.  w )  =  w )
1715, 16ax-mp 5 . . . . . 6  |-  ran  ( { w }  X.  w )  =  w
18 rn0 5104 . . . . . 6  |-  ran  (/)  =  (/)
1914, 17, 183eqtr3g 2518 . . . . 5  |-  ( ( { w }  X.  w )  =  (/)  ->  w  =  (/) )
2013, 19impbii 192 . . . 4  |-  ( w  =  (/)  <->  ( { w }  X.  w )  =  (/) )
2120necon3bii 2687 . . 3  |-  ( w  =/=  (/)  <->  ( { w }  X.  w )  =/=  (/) )
22 df-rex 2754 . . . 4  |-  ( E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t )  <->  E. t
( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
23 rneq 5078 . . . . . . . . . 10  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  ran  ( { w }  X.  w )  =  ran  ( { t }  X.  t ) )
24 vex 3059 . . . . . . . . . . . 12  |-  t  e. 
_V
2524snnz 4102 . . . . . . . . . . 11  |-  { t }  =/=  (/)
26 rnxp 5285 . . . . . . . . . . 11  |-  ( { t }  =/=  (/)  ->  ran  ( { t }  X.  t )  =  t )
2725, 26ax-mp 5 . . . . . . . . . 10  |-  ran  ( { t }  X.  t )  =  t
2823, 17, 273eqtr3g 2518 . . . . . . . . 9  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  ->  w  =  t )
29 sneq 3989 . . . . . . . . . . 11  |-  ( w  =  t  ->  { w }  =  { t } )
3029xpeq1d 4875 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  w ) )
31 xpeq2 4867 . . . . . . . . . 10  |-  ( w  =  t  ->  ( { t }  X.  w )  =  ( { t }  X.  t ) )
3230, 31eqtrd 2495 . . . . . . . . 9  |-  ( w  =  t  ->  ( { w }  X.  w )  =  ( { t }  X.  t ) )
3328, 32impbii 192 . . . . . . . 8  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  <->  w  =  t )
34 equcom 1872 . . . . . . . 8  |-  ( w  =  t  <->  t  =  w )
3533, 34bitri 257 . . . . . . 7  |-  ( ( { w }  X.  w )  =  ( { t }  X.  t )  <->  t  =  w )
3635anbi2i 705 . . . . . 6  |-  ( ( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) )  <->  ( t  e.  h  /\  t  =  w ) )
37 ancom 456 . . . . . 6  |-  ( ( t  e.  h  /\  t  =  w )  <->  ( t  =  w  /\  t  e.  h )
)
3836, 37bitri 257 . . . . 5  |-  ( ( t  e.  h  /\  ( { w }  X.  w )  =  ( { t }  X.  t ) )  <->  ( t  =  w  /\  t  e.  h ) )
3938exbii 1728 . . . 4  |-  ( E. t ( t  e.  h  /\  ( { w }  X.  w
)  =  ( { t }  X.  t
) )  <->  E. t
( t  =  w  /\  t  e.  h
) )
40 elequ1 1904 . . . . 5  |-  ( t  =  w  ->  (
t  e.  h  <->  w  e.  h ) )
412, 40ceqsexv 3095 . . . 4  |-  ( E. t ( t  =  w  /\  t  e.  h )  <->  w  e.  h )
4222, 39, 413bitrri 280 . . 3  |-  ( w  e.  h  <->  E. t  e.  h  ( {
w }  X.  w
)  =  ( { t }  X.  t
) )
4321, 42anbi12i 708 . 2  |-  ( ( w  =/=  (/)  /\  w  e.  h )  <->  ( ( { w }  X.  w )  =/=  (/)  /\  E. t  e.  h  ( { w }  X.  w )  =  ( { t }  X.  t ) ) )
448, 10, 433bitr4i 285 1  |-  ( ( { w }  X.  w )  e.  A  <->  ( w  =/=  (/)  /\  w  e.  h ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1454   E.wex 1673    e. wcel 1897   {cab 2447    =/= wne 2632   E.wrex 2749   (/)c0 3742   {csn 3979    X. cxp 4850   ran crn 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-xp 4858  df-rel 4859  df-cnv 4860  df-dm 4862  df-rn 4863
This theorem is referenced by:  dfac5lem5  8583
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