Users' Mathboxes Mathbox for Rodolfo Medina < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prtlem10 Unicode version

Theorem prtlem10 26733
Description: Lemma for prter3 26750. (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 6680 . . . . 5  |-  ( (  .~  Er  A  /\  z  e.  A )  ->  z  .~  z )
4 breq1 4026 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  z  <->  z  .~  z ) )
5 breq1 4026 . . . . . . . 8  |-  ( v  =  z  ->  (
v  .~  w  <->  z  .~  w ) )
64, 5anbi12d 691 . . . . . . 7  |-  ( v  =  z  ->  (
( v  .~  z  /\  v  .~  w
)  <->  ( z  .~  z  /\  z  .~  w
) ) )
76rspcev 2884 . . . . . 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 6678 . . . . . 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 2667 . . . 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 2791 . . . . . 6  |-  z  e. 
_V
18 vex 2791 . . . . . 6  |-  v  e. 
_V
1917, 18elec 6699 . . . . 5  |-  ( z  e.  [ v ]  .~  <->  v  .~  z
)
20 vex 2791 . . . . . 6  |-  w  e. 
_V
2120, 18elec 6699 . . . . 5  |-  ( w  e.  [ v ]  .~  <->  v  .~  w
)
2219, 21anbi12i 678 . . . 4  |-  ( ( z  e.  [ v ]  .~  /\  w  e.  [ v ]  .~  ) 
<->  ( v  .~  z  /\  v  .~  w
) )
2322rexbii 2568 . . 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 1623    e. wcel 1684   E.wrex 2544   class class class wbr 4023    Er wer 6657   [cec 6658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-er 6660  df-ec 6662
  Copyright terms: Public domain W3C validator