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Theorem prtlem10 26836
Description: Lemma for prter3 26853. (Contributed by Rodolfo Medina, 14-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
prtlem10  |-  (  .~  Er  A  ->  ( z  e.  A  ->  (
z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) ) ) )
Distinct variable groups:    w, v    z, v    v, A    v,  .~
Allowed substitution hints:    A( z, w)    .~ ( z, w)

Proof of Theorem prtlem10
StepHypRef Expression
1 simpr 447 . . . . 5  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  z  e.  A )
2 simpl 443 . . . . . 6  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  .~  Er  A )
32, 1erref 6696 . . . . 5  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  z  .~  z )
4 breq1 4042 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  z  <->  z  .~  z ) )
5 breq1 4042 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  w  <->  z  .~  w ) )
64, 5anbi12d 691 . . . . . . 7  |-  ( v  =  z  ->  (
( v  .~  z  /\  v  .~  w
)  <->  ( z  .~  z  /\  z  .~  w
) ) )
76rspcev 2897 . . . . . 6  |-  ( ( z  e.  A  /\  ( z  .~  z  /\  z  .~  w
) )  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w ) )
87expr 598 . . . . 5  |-  ( ( z  e.  A  /\  z  .~  z )  -> 
( z  .~  w  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) ) )
91, 3, 8syl2anc 642 . . . 4  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) ) )
10 simplll 734 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  .~  Er  A
)
11 simprl 732 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  v  .~  z
)
12 simprr 733 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  v  .~  w
)
1310, 11, 12ertr3d 6694 . . . . . 6  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  z  .~  w
)
1413ex 423 . . . . 5  |-  ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  ->  (
( v  .~  z  /\  v  .~  w
)  ->  z  .~  w ) )
1514rexlimdva 2680 . . . 4  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( E. v  e.  A  ( v  .~  z  /\  v  .~  w
)  ->  z  .~  w ) )
169, 15impbid 183 . . 3  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  <->  E. v  e.  A  ( v  .~  z  /\  v  .~  w ) ) )
17 vex 2804 . . . . . 6  |-  z  e. 
_V
18 vex 2804 . . . . . 6  |-  v  e. 
_V
1917, 18elec 6715 . . . . 5  |-  ( z  e.  [ v ]  .~  <->  v  .~  z
)
20 vex 2804 . . . . . 6  |-  w  e. 
_V
2120, 18elec 6715 . . . . 5  |-  ( w  e.  [ v ]  .~  <->  v  .~  w
)
2219, 21anbi12i 678 . . . 4  |-  ( ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) 
<->  ( v  .~  z  /\  v  .~  w
) )
2322rexbii 2581 . . 3  |-  ( E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) 
<->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) )
2416, 23syl6bbr 254 . 2  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) ) )
2524ex 423 1  |-  (  .~  Er  A  ->  ( z  e.  A  ->  (
z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039    Er wer 6673   [cec 6674
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-er 6676  df-ec 6678
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