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Theorem wfrlem13 7059
Description: Lemma for well-founded recursion. From here through wfrlem16 7062, 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 7057 . . . . 5  |-  Fun  F
5 vex 3083 . . . . . 6  |-  z  e. 
_V
6 fvex 5891 . . . . . 6  |-  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) )  e.  _V
75, 6funsn 5649 . . . . 5  |-  Fun  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. }
84, 7pm3.2i 456 . . . 4  |-  ( Fun 
F  /\  Fun  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )
96dmsnop 5329 . . . . . 6  |-  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. }  =  {
z }
109ineq2i 3661 . . . . 5  |-  ( dom 
F  i^i  dom  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  i^i  {
z } )
11 eldifn 3588 . . . . . 6  |-  ( z  e.  ( A  \  dom  F )  ->  -.  z  e.  dom  F )
12 disjsn 4060 . . . . . 6  |-  ( ( dom  F  i^i  {
z } )  =  (/) 
<->  -.  z  e.  dom  F )
1311, 12sylibr 215 . . . . 5  |-  ( z  e.  ( A  \  dom  F )  ->  ( dom  F  i^i  { z } )  =  (/) )
1410, 13syl5eq 2475 . . . 4  |-  ( z  e.  ( A  \  dom  F )  ->  ( dom  F  i^i  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  (/) )
15 funun 5643 . . . 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 667 . . 3  |-  ( z  e.  ( A  \  dom  F )  ->  Fun  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } ) )
17 dmun 5060 . . . 4  |-  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  dom  {
<. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
189uneq2i 3617 . . . 4  |-  ( dom 
F  u.  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  ( dom  F  u.  { z } )
1917, 18eqtri 2451 . . 3  |-  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  {
z } )
2016, 19jctir 540 . 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 5688 . . 3  |-  ( C  Fn  ( dom  F  u.  { z } )  <-> 
( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  Fn  ( dom  F  u.  { z } ) )
23 df-fn 5604 . . 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 252 . 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 215 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 370    = wceq 1437    e. wcel 1872    \ cdif 3433    u. cun 3434    i^i cin 3435   (/)c0 3761   {csn 3998   <.cop 4004   Se wse 4810    We wwe 4811   dom cdm 4853    |` cres 4855   Predcpred 5398   Fun wfun 5595    Fn wfn 5596   ` cfv 5601  wrecscwrecs 7038
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-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660
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 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-wrecs 7039
This theorem is referenced by:  wfrlem14  7060  wfrlem15  7061
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