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Theorem prtlem10 26714
Description: Lemma for prter3 26731. (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 448 . . . . 5  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  z  e.  A )
2 simpl 444 . . . . . 6  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  .~  Er  A )
32, 1erref 6925 . . . . 5  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  z  .~  z )
4 breq1 4215 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  z  <->  z  .~  z ) )
5 breq1 4215 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  w  <->  z  .~  w ) )
64, 5anbi12d 692 . . . . . . 7  |-  ( v  =  z  ->  (
( v  .~  z  /\  v  .~  w
)  <->  ( z  .~  z  /\  z  .~  w
) ) )
76rspcev 3052 . . . . . 6  |-  ( ( z  e.  A  /\  ( z  .~  z  /\  z  .~  w
) )  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w ) )
87expr 599 . . . . 5  |-  ( ( z  e.  A  /\  z  .~  z )  -> 
( z  .~  w  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) ) )
91, 3, 8syl2anc 643 . . . 4  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) ) )
10 simplll 735 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  .~  Er  A
)
11 simprl 733 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  v  .~  z
)
12 simprr 734 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  v  .~  w
)
1310, 11, 12ertr3d 6923 . . . . . 6  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  z  .~  w
)
1413ex 424 . . . . 5  |-  ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  ->  (
( v  .~  z  /\  v  .~  w
)  ->  z  .~  w ) )
1514rexlimdva 2830 . . . 4  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( E. v  e.  A  ( v  .~  z  /\  v  .~  w
)  ->  z  .~  w ) )
169, 15impbid 184 . . 3  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  <->  E. v  e.  A  ( v  .~  z  /\  v  .~  w ) ) )
17 vex 2959 . . . . . 6  |-  z  e. 
_V
18 vex 2959 . . . . . 6  |-  v  e. 
_V
1917, 18elec 6944 . . . . 5  |-  ( z  e.  [ v ]  .~  <->  v  .~  z
)
20 vex 2959 . . . . . 6  |-  w  e. 
_V
2120, 18elec 6944 . . . . 5  |-  ( w  e.  [ v ]  .~  <->  v  .~  w
)
2219, 21anbi12i 679 . . . 4  |-  ( ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) 
<->  ( v  .~  z  /\  v  .~  w
) )
2322rexbii 2730 . . 3  |-  ( E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) 
<->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) )
2416, 23syl6bbr 255 . 2  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) ) )
2524ex 424 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 177    /\ wa 359    e. wcel 1725   E.wrex 2706   class class class wbr 4212    Er wer 6902   [cec 6903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-er 6905  df-ec 6907
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