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Theorem wfrlem10 7000
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 6998 . . 3  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  z )  =  (/)  <->  Pred ( R ,  A ,  z )  =  Pred ( R ,  dom  F ,  z ) )
32biimpi 197 . 2  |-  ( Pred ( R ,  ( A  \  dom  F
) ,  z )  =  (/)  ->  Pred ( R ,  A , 
z )  =  Pred ( R ,  dom  F ,  z ) )
4 predss 5349 . . . 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 462 . . . . . 6  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  w  e.  dom  F )
7 eldifn 3531 . . . . . . . . . 10  |-  ( z  e.  ( A  \  dom  F )  ->  -.  z  e.  dom  F )
8 eleq1 2494 . . . . . . . . . . 11  |-  ( w  =  z  ->  (
w  e.  dom  F  <->  z  e.  dom  F ) )
98notbid 295 . . . . . . . . . 10  |-  ( w  =  z  ->  ( -.  w  e.  dom  F  <->  -.  z  e.  dom  F ) )
107, 9syl5ibrcom 225 . . . . . . . . 9  |-  ( z  e.  ( A  \  dom  F )  ->  (
w  =  z  ->  -.  w  e.  dom  F ) )
1110con2d 118 . . . . . . . 8  |-  ( z  e.  ( A  \  dom  F )  ->  (
w  e.  dom  F  ->  -.  w  =  z ) )
1211imp 430 . . . . . . 7  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  -.  w  =  z )
131wfrdmcl 6999 . . . . . . . . . 10  |-  ( w  e.  dom  F  ->  Pred ( R ,  A ,  w )  C_  dom  F )
1413adantl 467 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  Pred ( R ,  A ,  w
)  C_  dom  F )
15 ssel 3401 . . . . . . . . . . . 12  |-  ( Pred ( R ,  A ,  w )  C_  dom  F  ->  ( z  e. 
Pred ( R ,  A ,  w )  ->  z  e.  dom  F
) )
1615con3d 138 . . . . . . . . . . 11  |-  ( Pred ( R ,  A ,  w )  C_  dom  F  ->  ( -.  z  e.  dom  F  ->  -.  z  e.  Pred ( R ,  A ,  w
) ) )
177, 16syl5com 31 . . . . . . . . . 10  |-  ( z  e.  ( A  \  dom  F )  ->  ( Pred ( R ,  A ,  w )  C_  dom  F  ->  -.  z  e.  Pred ( R ,  A ,  w ) ) )
1817adantr 466 . . . . . . . . 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 3530 . . . . . . . . 9  |-  ( z  e.  ( A  \  dom  F )  ->  z  e.  A )
21 elpredg 5356 . . . . . . . . . 10  |-  ( ( w  e.  dom  F  /\  z  e.  A
)  ->  ( z  e.  Pred ( R ,  A ,  w )  <->  z R w ) )
2221ancoms 454 . . . . . . . . 9  |-  ( ( z  e.  A  /\  w  e.  dom  F )  ->  ( z  e. 
Pred ( R ,  A ,  w )  <->  z R w ) )
2320, 22sylan 473 . . . . . . . 8  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  ( z  e.  Pred ( R ,  A ,  w )  <->  z R w ) )
2419, 23mtbid 301 . . . . . . 7  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  -.  z R w )
251wfrdmss 6997 . . . . . . . . 9  |-  dom  F  C_  A
2625sseli 3403 . . . . . . . 8  |-  ( w  e.  dom  F  ->  w  e.  A )
27 wfrlem10.1 . . . . . . . . . 10  |-  R  We  A
28 weso 4787 . . . . . . . . . 10  |-  ( R  We  A  ->  R  Or  A )
2927, 28ax-mp 5 . . . . . . . . 9  |-  R  Or  A
30 solin 4740 . . . . . . . . 9  |-  ( ( R  Or  A  /\  ( w  e.  A  /\  z  e.  A
) )  ->  (
w R z  \/  w  =  z  \/  z R w ) )
3129, 30mpan 674 . . . . . . . 8  |-  ( ( w  e.  A  /\  z  e.  A )  ->  ( w R z  \/  w  =  z  \/  z R w ) )
3226, 20, 31syl2anr 480 . . . . . . 7  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  ( w R z  \/  w  =  z  \/  z R w ) )
3312, 24, 32ecase23d 1368 . . . . . 6  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  w R
z )
34 vex 3025 . . . . . . 7  |-  z  e. 
_V
35 vex 3025 . . . . . . . 8  |-  w  e. 
_V
3635elpred 5355 . . . . . . 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 668 . . . . 5  |-  ( ( z  e.  ( A 
\  dom  F )  /\  w  e.  dom  F )  ->  w  e.  Pred ( R ,  dom  F ,  z ) )
3938ex 435 . . . 4  |-  ( z  e.  ( A  \  dom  F )  ->  (
w  e.  dom  F  ->  w  e.  Pred ( R ,  dom  F , 
z ) ) )
4039ssrdv 3413 . . 3  |-  ( z  e.  ( A  \  dom  F )  ->  dom  F 
C_  Pred ( R ,  dom  F ,  z ) )
415, 40eqssd 3424 . 2  |-  ( z  e.  ( A  \  dom  F )  ->  Pred ( R ,  dom  F , 
z )  =  dom  F )
423, 41sylan9eqr 2484 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 187    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1872   _Vcvv 3022    \ cdif 3376    C_ wss 3379   (/)c0 3704   class class class wbr 4366    Or wor 4716    We wwe 4754   dom cdm 4796   Predcpred 5341  wrecscwrecs 6982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-so 4718  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552  df-wrecs 6983
This theorem is referenced by:  wfrlem15  7005
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