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Theorem prtlem10 29015
Description: Lemma for prter3 29032. (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 461 . . . . 5  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  z  e.  A )
2 simpl 457 . . . . . 6  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  .~  Er  A )
32, 1erref 7126 . . . . 5  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  z  .~  z )
4 breq1 4300 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  z  <->  z  .~  z ) )
5 breq1 4300 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  w  <->  z  .~  w ) )
64, 5anbi12d 710 . . . . . . 7  |-  ( v  =  z  ->  (
( v  .~  z  /\  v  .~  w
)  <->  ( z  .~  z  /\  z  .~  w
) ) )
76rspcev 3078 . . . . . 6  |-  ( ( z  e.  A  /\  ( z  .~  z  /\  z  .~  w
) )  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w ) )
87expr 615 . . . . 5  |-  ( ( z  e.  A  /\  z  .~  z )  -> 
( z  .~  w  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) ) )
91, 3, 8syl2anc 661 . . . 4  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  ->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) ) )
10 simplll 757 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  .~  Er  A
)
11 simprl 755 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  v  .~  z
)
12 simprr 756 . . . . . . 7  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  v  .~  w
)
1310, 11, 12ertr3d 7124 . . . . . 6  |-  ( ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  /\  (
v  .~  z  /\  v  .~  w ) )  ->  z  .~  w
)
1413ex 434 . . . . 5  |-  ( ( (  .~  Er  A  /\  z  e.  A
)  /\  v  e.  A )  ->  (
( v  .~  z  /\  v  .~  w
)  ->  z  .~  w ) )
1514rexlimdva 2846 . . . 4  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( E. v  e.  A  ( v  .~  z  /\  v  .~  w
)  ->  z  .~  w ) )
169, 15impbid 191 . . 3  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  <->  E. v  e.  A  ( v  .~  z  /\  v  .~  w ) ) )
17 vex 2980 . . . . . 6  |-  z  e. 
_V
18 vex 2980 . . . . . 6  |-  v  e. 
_V
1917, 18elec 7145 . . . . 5  |-  ( z  e.  [ v ]  .~  <->  v  .~  z
)
20 vex 2980 . . . . . 6  |-  w  e. 
_V
2120, 18elec 7145 . . . . 5  |-  ( w  e.  [ v ]  .~  <->  v  .~  w
)
2219, 21anbi12i 697 . . . 4  |-  ( ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) 
<->  ( v  .~  z  /\  v  .~  w
) )
2322rexbii 2745 . . 3  |-  ( E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) 
<->  E. v  e.  A  ( v  .~  z  /\  v  .~  w
) )
2416, 23syl6bbr 263 . 2  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  ( z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) ) )
2524ex 434 1  |-  (  .~  Er  A  ->  ( z  e.  A  ->  (
z  .~  w  <->  E. v  e.  A  ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1756   E.wrex 2721   class class class wbr 4297    Er wer 7103   [cec 7104
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-er 7106  df-ec 7108
This theorem is referenced by: (None)
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