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Theorem wfrlem13 7032
Description: Lemma for well-founded recursion. From here through wfrlem16 7035, 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, 3wfrfun 7030 . . . . 5  |-  Fun  F
5 vex 3061 . . . . . 6  |-  z  e. 
_V
6 fvex 5858 . . . . . 6  |-  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) )  e.  _V
75, 6funsn 5616 . . . . 5  |-  Fun  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. }
84, 7pm3.2i 453 . . . 4  |-  ( Fun 
F  /\  Fun  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )
96dmsnop 5297 . . . . . 6  |-  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. }  =  {
z }
109ineq2i 3637 . . . . 5  |-  ( dom 
F  i^i  dom  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  i^i  {
z } )
11 eldifn 3565 . . . . . 6  |-  ( z  e.  ( A  \  dom  F )  ->  -.  z  e.  dom  F )
12 disjsn 4031 . . . . . 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 2455 . . . 4  |-  ( z  e.  ( A  \  dom  F )  ->  ( dom  F  i^i  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  (/) )
15 funun 5610 . . . 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 661 . . 3  |-  ( z  e.  ( A  \  dom  F )  ->  Fun  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } ) )
17 dmun 5029 . . . 4  |-  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  dom  {
<. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
189uneq2i 3593 . . . 4  |-  ( dom 
F  u.  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  ( dom  F  u.  { z } )
1917, 18eqtri 2431 . . 3  |-  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  {
z } )
2016, 19jctir 536 . 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 5655 . . 3  |-  ( C  Fn  ( dom  F  u.  { z } )  <-> 
( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  Fn  ( dom  F  u.  { z } ) )
23 df-fn 5571 . . 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 367    = wceq 1405    e. wcel 1842    \ cdif 3410    u. cun 3411    i^i cin 3412   (/)c0 3737   {csn 3971   <.cop 3977   Se wse 4779    We wwe 4780   dom cdm 4822    |` cres 4824   Predcpred 5365   Fun wfun 5562    Fn wfn 5563   ` cfv 5568  wrecscwrecs 7011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-se 4782  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-iota 5532  df-fun 5570  df-fn 5571  df-fv 5576  df-wrecs 7012
This theorem is referenced by:  wfrlem14  7033  wfrlem15  7034
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