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Theorem wfrlem10 27853
Description: Lemma for well-founded recursion. When  z is an  R minimal element of  ( A  \  dom  F ), then its predecessor class is equal to  dom  F. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem10.1  |-  R  We  A
wfrlem10.2  |-  F  = wrecs ( R ,  A ,  G )
Assertion
Ref Expression
wfrlem10  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  Pred ( R ,  A , 
z )  =  dom  F )
Distinct variable group:    z, A
Allowed substitution hints:    R( z)    F( z)    G( z)

Proof of Theorem wfrlem10
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 wfrlem10.2 . . . 4  |-  F  = wrecs ( R ,  A ,  G )
21wfrlem8 27851 . . 3  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  z )  =  (/)  <->  Pred ( R ,  A ,  z )  =  Pred ( R ,  dom  F ,  z ) )
32biimpi 194 . 2  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  z )  =  (/)  ->  Pred ( R ,  A , 
z )  =  Pred ( R ,  dom  F ,  z ) )
4 predss 27752 . . . 4  |-  Pred ( R ,  dom  F , 
z )  C_  dom  F
54a1i 11 . . 3  |-  ( z  e.  ( A  \  dom  F )  ->  Pred ( R ,  dom  F , 
z )  C_  dom  F )
6 simpr 461 . . . . . 6  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  w  e.  dom  F )
7 eldifn 3563 . . . . . . . . . 10  |-  ( z  e.  ( A  \  dom  F )  ->  -.  z  e.  dom  F )
8 eleq1 2520 . . . . . . . . . . 11  |-  ( w  =  z  ->  (
w  e.  dom  F  <->  z  e.  dom  F ) )
98notbid 294 . . . . . . . . . 10  |-  ( w  =  z  ->  ( -.  w  e.  dom  F  <->  -.  z  e.  dom  F ) )
107, 9syl5ibrcom 222 . . . . . . . . 9  |-  ( z  e.  ( A  \  dom  F )  ->  (
w  =  z  ->  -.  w  e.  dom  F ) )
1110con2d 115 . . . . . . . 8  |-  ( z  e.  ( A  \  dom  F )  ->  (
w  e.  dom  F  ->  -.  w  =  z ) )
1211imp 429 . . . . . . 7  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  -.  w  =  z )
131wfrlem9 27852 . . . . . . . . . 10  |-  ( w  e.  dom  F  ->  Pred ( R ,  A ,  w )  C_  dom  F )
1413adantl 466 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  Pred ( R ,  A ,  w
)  C_  dom  F )
15 ssel 3434 . . . . . . . . . . . 12  |-  ( Pred ( R ,  A ,  w )  C_  dom  F  ->  ( z  e. 
Pred ( R ,  A ,  w )  ->  z  e.  dom  F
) )
1615con3d 133 . . . . . . . . . . 11  |-  ( Pred ( R ,  A ,  w )  C_  dom  F  ->  ( -.  z  e.  dom  F  ->  -.  z  e.  Pred ( R ,  A ,  w
) ) )
177, 16syl5com 30 . . . . . . . . . 10  |-  ( z  e.  ( A  \  dom  F )  ->  ( Pred ( R ,  A ,  w )  C_  dom  F  ->  -.  z  e.  Pred ( R ,  A ,  w ) ) )
1817adantr 465 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  ( Pred ( R ,  A ,  w )  C_  dom  F  ->  -.  z  e.  Pred ( R ,  A ,  w ) ) )
1914, 18mpd 15 . . . . . . . 8  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  -.  z  e.  Pred ( R ,  A ,  w )
)
20 eldifi 3562 . . . . . . . . 9  |-  ( z  e.  ( A  \  dom  F )  ->  z  e.  A )
21 elpredg 27759 . . . . . . . . . 10  |-  ( ( w  e.  dom  F  /\  z  e.  A
)  ->  ( z  e.  Pred ( R ,  A ,  w )  <->  z R w ) )
2221ancoms 453 . . . . . . . . 9  |-  ( ( z  e.  A  /\  w  e.  dom  F )  ->  ( z  e. 
Pred ( R ,  A ,  w )  <->  z R w ) )
2320, 22sylan 471 . . . . . . . 8  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  ( z  e.  Pred ( R ,  A ,  w )  <->  z R w ) )
2419, 23mtbid 300 . . . . . . 7  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  -.  z R w )
251wfrlem7 27850 . . . . . . . . 9  |-  dom  F  C_  A
2625sseli 3436 . . . . . . . 8  |-  ( w  e.  dom  F  ->  w  e.  A )
27 wfrlem10.1 . . . . . . . . . 10  |-  R  We  A
28 weso 4795 . . . . . . . . . 10  |-  ( R  We  A  ->  R  Or  A )
2927, 28ax-mp 5 . . . . . . . . 9  |-  R  Or  A
30 solin 4748 . . . . . . . . 9  |-  ( ( R  Or  A  /\  ( w  e.  A  /\  z  e.  A
) )  ->  (
w R z  \/  w  =  z  \/  z R w ) )
3129, 30mpan 670 . . . . . . . 8  |-  ( ( w  e.  A  /\  z  e.  A )  ->  ( w R z  \/  w  =  z  \/  z R w ) )
3226, 20, 31syl2anr 478 . . . . . . 7  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  ( w R z  \/  w  =  z  \/  z R w ) )
3312, 24, 32ecase23d 1323 . . . . . 6  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  w R
z )
34 vex 3057 . . . . . . 7  |-  z  e. 
_V
35 vex 3057 . . . . . . . 8  |-  w  e. 
_V
3635elpred 27758 . . . . . . 7  |-  ( z  e.  _V  ->  (
w  e.  Pred ( R ,  dom  F , 
z )  <->  ( w  e.  dom  F  /\  w R z ) ) )
3734, 36ax-mp 5 . . . . . 6  |-  ( w  e.  Pred ( R ,  dom  F ,  z )  <-> 
( w  e.  dom  F  /\  w R z ) )
386, 33, 37sylanbrc 664 . . . . 5  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  w  e.  Pred ( R ,  dom  F ,  z ) )
3938ex 434 . . . 4  |-  ( z  e.  ( A  \  dom  F )  ->  (
w  e.  dom  F  ->  w  e.  Pred ( R ,  dom  F , 
z ) ) )
4039ssrdv 3446 . . 3  |-  ( z  e.  ( A  \  dom  F )  ->  dom  F 
C_  Pred ( R ,  dom  F ,  z ) )
415, 40eqssd 3457 . 2  |-  ( z  e.  ( A  \  dom  F )  ->  Pred ( R ,  dom  F , 
z )  =  dom  F )
423, 41sylan9eqr 2512 1  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  Pred ( R ,  A , 
z )  =  dom  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1757   _Vcvv 3054    \ cdif 3409    C_ wss 3412   (/)c0 3721   class class class wbr 4376    Or wor 4724    We wwe 4762   dom cdm 4924   Predcpred 27744  wrecscwrecs 27836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pr 4615
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-so 4726  df-we 4765  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-fv 5510  df-pred 27745  df-wrecs 27837
This theorem is referenced by:  wfrlem15  27858
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