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Theorem wfrlem16 25485
Description: Lemma for well-founded recursion. If  z is  R minimal in  ( A  \  dom  F ), then  C is acceptable and thus a subset of  F, but  dom  C is bigger than  dom  F. Thus, 
z cannot be minimal, so  ( A  \  dom  F ) must be empty, and (due to wfrlem7 25476),  dom  F  =  A. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypotheses
Ref Expression
wfrlem13.1  |-  R  We  A
wfrlem13.2  |-  R Se  A
wfrlem13.3  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x
)  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
wfrlem13.4  |-  F  = 
U. B
wfrlem13.5  |-  C  =  ( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
Assertion
Ref Expression
wfrlem16  |-  dom  F  =  A
Distinct variable groups:    A, f, x, y, z    f, F, x, y, z    f, G, x, y    R, f, x, y, z    C, f, x, y
Allowed substitution hints:    B( x, y, z, f)    C( z)    G( z)

Proof of Theorem wfrlem16
StepHypRef Expression
1 wfrlem13.3 . . 3  |-  B  =  { f  |  E. x ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x
)  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }
2 wfrlem13.4 . . 3  |-  F  = 
U. B
31, 2wfrlem7 25476 . 2  |-  dom  F  C_  A
4 eldifn 3430 . . . . . 6  |-  ( z  e.  ( A  \  dom  F )  ->  -.  z  e.  dom  F )
5 ssun2 3471 . . . . . . . . 9  |-  { z }  C_  ( dom  F  u.  { z } )
6 vex 2919 . . . . . . . . . 10  |-  z  e. 
_V
76snid 3801 . . . . . . . . 9  |-  z  e. 
{ z }
85, 7sselii 3305 . . . . . . . 8  |-  z  e.  ( dom  F  u.  { z } )
9 wfrlem13.5 . . . . . . . . . 10  |-  C  =  ( F  u.  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
109dmeqi 5030 . . . . . . . . 9  |-  dom  C  =  dom  ( F  u.  {
<. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
11 dmun 5035 . . . . . . . . 9  |-  dom  ( F  u.  { <. z ,  ( G `  ( F  |`  Pred ( R ,  A , 
z ) ) )
>. } )  =  ( dom  F  u.  dom  {
<. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )
12 fvex 5701 . . . . . . . . . . 11  |-  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) )  e.  _V
1312dmsnop 5303 . . . . . . . . . 10  |-  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. }  =  {
z }
1413uneq2i 3458 . . . . . . . . 9  |-  ( dom 
F  u.  dom  { <. z ,  ( G `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) >. } )  =  ( dom  F  u.  { z } )
1510, 11, 143eqtri 2428 . . . . . . . 8  |-  dom  C  =  ( dom  F  u.  { z } )
168, 15eleqtrri 2477 . . . . . . 7  |-  z  e. 
dom  C
17 wfrlem13.1 . . . . . . . . . . . 12  |-  R  We  A
18 wfrlem13.2 . . . . . . . . . . . 12  |-  R Se  A
1917, 18, 1, 2, 9wfrlem15 25484 . . . . . . . . . . 11  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  C  e.  B )
20 elssuni 4003 . . . . . . . . . . 11  |-  ( C  e.  B  ->  C  C_ 
U. B )
2119, 20syl 16 . . . . . . . . . 10  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  C  C_ 
U. B )
2221, 2syl6sseqr 3355 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  C  C_  F )
23 dmss 5028 . . . . . . . . 9  |-  ( C 
C_  F  ->  dom  C 
C_  dom  F )
2422, 23syl 16 . . . . . . . 8  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  dom  C 
C_  dom  F )
2524sseld 3307 . . . . . . 7  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  (
z  e.  dom  C  ->  z  e.  dom  F
) )
2616, 25mpi 17 . . . . . 6  |-  ( ( z  e.  ( A 
\  dom  F )  /\  Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )  ->  z  e.  dom  F )
274, 26mtand 641 . . . . 5  |-  ( z  e.  ( A  \  dom  F )  ->  -.  Pred ( R ,  ( A  \  dom  F
) ,  z )  =  (/) )
2827nrex 2768 . . . 4  |-  -.  E. z  e.  ( A  \  dom  F ) Pred ( R ,  ( A  \  dom  F
) ,  z )  =  (/)
29 df-ne 2569 . . . . 5  |-  ( ( A  \  dom  F
)  =/=  (/)  <->  -.  ( A  \  dom  F )  =  (/) )
30 difss 3434 . . . . . 6  |-  ( A 
\  dom  F )  C_  A
3117, 18tz6.26i 25420 . . . . . 6  |-  ( ( ( A  \  dom  F )  C_  A  /\  ( A  \  dom  F
)  =/=  (/) )  ->  E. z  e.  ( A  \  dom  F )
Pred ( R , 
( A  \  dom  F ) ,  z )  =  (/) )
3230, 31mpan 652 . . . . 5  |-  ( ( A  \  dom  F
)  =/=  (/)  ->  E. z  e.  ( A  \  dom  F ) Pred ( R ,  ( A  \  dom  F ) ,  z )  =  (/) )
3329, 32sylbir 205 . . . 4  |-  ( -.  ( A  \  dom  F )  =  (/)  ->  E. z  e.  ( A  \  dom  F ) Pred ( R ,  ( A  \  dom  F ) ,  z )  =  (/) )
3428, 33mt3 173 . . 3  |-  ( A 
\  dom  F )  =  (/)
35 ssdif0 3646 . . 3  |-  ( A 
C_  dom  F  <->  ( A  \  dom  F )  =  (/) )
3634, 35mpbir 201 . 2  |-  A  C_  dom  F
373, 36eqssi 3324 1  |-  dom  F  =  A
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667    \ cdif 3277    u. cun 3278    C_ wss 3280   (/)c0 3588   {csn 3774   <.cop 3777   U.cuni 3975   Se wse 4499    We wwe 4500   dom cdm 4837    |` cres 4839    Fn wfn 5408   ` cfv 5413   Predcpred 25381
This theorem is referenced by:  wfr1  25486  wfr2  25487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-pred 25382
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