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Theorem wfrlem13 28932
Description: Lemma for well-founded recursion. From here through wfrlem16 28935, we aim to prove that  dom  F  =  A. We do this by supposing that there is an element  z of  A that is not in  dom  F. We then define  C by extending  dom  F with the appropriate value at  z. We then show that  z cannot be an  R minimal element of  ( A  \  dom  F ), meaning that  ( A  \  dom  F ) must be empty, so  dom  F  =  A. Here, we show that  C is a function extending the domain of  F by one. (Contributed by Scott Fenton, 21-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
wfrlem13.1  |-  R  We  A
wfrlem13.2  |-  R Se  A
wfrlem13.3  |-  F  = wrecs ( R ,  A ,  G )
wfrlem13.4  |-  C  =  ( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
Assertion
Ref Expression
wfrlem13  |-  ( z  e.  ( A  \  dom  F )  ->  C  Fn  ( dom  F  u.  { z } ) )
Distinct variable groups:    z, A    z, F    z, R
Allowed substitution hints:    C( z)    G( z)

Proof of Theorem wfrlem13
StepHypRef Expression
1 wfrlem13.1 . . . . . 6  |-  R  We  A
2 wfrlem13.2 . . . . . 6  |-  R Se  A
3 wfrlem13.3 . . . . . 6  |-  F  = wrecs ( R ,  A ,  G )
41, 2, 3wfrlem11 28930 . . . . 5  |-  Fun  F
5 vex 3116 . . . . . 6  |-  z  e. 
_V
6 fvex 5874 . . . . . 6  |-  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) )  e.  _V
75, 6funsn 5634 . . . . 5  |-  Fun  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. }
84, 7pm3.2i 455 . . . 4  |-  ( Fun 
F  /\  Fun  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )
96dmsnop 5480 . . . . . 6  |-  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. }  =  {
z }
109ineq2i 3697 . . . . 5  |-  ( dom 
F  i^i  dom  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  i^i  {
z } )
11 eldifn 3627 . . . . . 6  |-  ( z  e.  ( A  \  dom  F )  ->  -.  z  e.  dom  F )
12 disjsn 4088 . . . . . 6  |-  ( ( dom  F  i^i  {
z } )  =  (/) 
<->  -.  z  e.  dom  F )
1311, 12sylibr 212 . . . . 5  |-  ( z  e.  ( A  \  dom  F )  ->  ( dom  F  i^i  { z } )  =  (/) )
1410, 13syl5eq 2520 . . . 4  |-  ( z  e.  ( A  \  dom  F )  ->  ( dom  F  i^i  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  (/) )
15 funun 5628 . . . 4  |-  ( ( ( Fun  F  /\  Fun  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
) ) >. } )  /\  ( dom  F  i^i  dom  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  (/) )  ->  Fun  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
) ) >. } ) )
168, 14, 15sylancr 663 . . 3  |-  ( z  e.  ( A  \  dom  F )  ->  Fun  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } ) )
17 dmun 5207 . . . 4  |-  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  dom  {
<. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
189uneq2i 3655 . . . 4  |-  ( dom 
F  u.  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  ( dom  F  u.  { z } )
1917, 18eqtri 2496 . . 3  |-  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  {
z } )
2016, 19jctir 538 . 2  |-  ( z  e.  ( A  \  dom  F )  ->  ( Fun  ( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  /\  dom  ( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  ( dom  F  u.  { z } ) ) )
21 wfrlem13.4 . . . 4  |-  C  =  ( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
2221fneq1i 5673 . . 3  |-  ( C  Fn  ( dom  F  u.  { z } )  <-> 
( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  Fn  ( dom  F  u.  { z } ) )
23 df-fn 5589 . . 3  |-  ( ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  Fn  ( dom  F  u.  { z } )  <->  ( Fun  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  /\  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  {
z } ) ) )
2422, 23bitri 249 . 2  |-  ( C  Fn  ( dom  F  u.  { z } )  <-> 
( Fun  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A ,  z )
) ) >. } )  /\  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  {
z } ) ) )
2520, 24sylibr 212 1  |-  ( z  e.  ( A  \  dom  F )  ->  C  Fn  ( dom  F  u.  { z } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   <.cop 4033   Se wse 4836    We wwe 4837   dom cdm 4999    |` cres 5001   Fun wfun 5580    Fn wfn 5581   ` cfv 5586   Predcpred 28820  wrecscwrecs 28912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-pred 28821  df-wrecs 28913
This theorem is referenced by:  wfrlem14  28933  wfrlem15  28934
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