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Theorem wfrlem9 25538
Description: Lemma for well-founded recursion. If  X  e.  dom  F, then its predecessor class is a subset of  dom  F. (Contributed by Scott Fenton, 21-Apr-2011.)
Hypothesis
Ref Expression
wfrlem6.1  |-  F  = wrecs ( R ,  A ,  G )
Assertion
Ref Expression
wfrlem9  |-  ( X  e.  dom  F  ->  Pred ( R ,  A ,  X )  C_  dom  F )

Proof of Theorem wfrlem9
Dummy variables  f 
g  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem6.1 . . . . . . . 8  |-  F  = wrecs ( R ,  A ,  G )
2 df-wrecs 25523 . . . . . . . 8  |- wrecs ( R ,  A ,  G
)  =  U. {
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 ) ) ) ) }
31, 2eqtri 2455 . . . . . . 7  |-  F  = 
U. { 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 )
) ) ) }
43dmeqi 5063 . . . . . 6  |-  dom  F  =  dom  U. { 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 )
) ) ) }
5 dmuni 5071 . . . . . 6  |-  dom  U. { 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 )
) ) ) }  =  U_ g  e. 
{ 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 )
) ) ) } dom  g
64, 5eqtri 2455 . . . . 5  |-  dom  F  =  U_ g  e.  {
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 ) ) ) ) } dom  g
76eleq2i 2499 . . . 4  |-  ( X  e.  dom  F  <->  X  e.  U_ g  e.  { 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 )
) ) ) } dom  g )
8 eliun 4089 . . . 4  |-  ( X  e.  U_ g  e. 
{ 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 )
) ) ) } dom  g  <->  E. g  e.  { 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 )
) ) ) } X  e.  dom  g
)
97, 8bitri 241 . . 3  |-  ( X  e.  dom  F  <->  E. g  e.  { 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 )
) ) ) } X  e.  dom  g
)
10 eqid 2435 . . . . . . . 8  |-  { 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 )
) ) ) }  =  { 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 )
) ) ) }
1110wfrlem1 25530 . . . . . . 7  |-  { 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 )
) ) ) }  =  { g  |  E. z ( g  Fn  z  /\  (
z  C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w )
) ) ) }
1211abeq2i 2542 . . . . . 6  |-  ( g  e.  { 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 )
) ) ) }  <->  E. z ( g  Fn  z  /\  ( z 
C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w )
) ) ) )
13 predeq3 25435 . . . . . . . . . . . . 13  |-  ( w  =  X  ->  Pred ( R ,  A ,  w )  =  Pred ( R ,  A ,  X ) )
1413sseq1d 3367 . . . . . . . . . . . 12  |-  ( w  =  X  ->  ( Pred ( R ,  A ,  w )  C_  z  <->  Pred ( R ,  A ,  X )  C_  z
) )
1514rspccv 3041 . . . . . . . . . . 11  |-  ( A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  ->  ( X  e.  z  ->  Pred ( R ,  A ,  X )  C_  z ) )
1615adantl 453 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)  ->  ( X  e.  z  ->  Pred ( R ,  A ,  X )  C_  z
) )
17 fndm 5536 . . . . . . . . . . . . 13  |-  ( g  Fn  z  ->  dom  g  =  z )
1817eleq2d 2502 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  ( X  e.  dom  g  <->  X  e.  z ) )
1917sseq2d 3368 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  ( Pred ( R ,  A ,  X )  C_  dom  g 
<-> 
Pred ( R ,  A ,  X )  C_  z ) )
2018, 19imbi12d 312 . . . . . . . . . . 11  |-  ( g  Fn  z  ->  (
( X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_ 
dom  g )  <->  ( X  e.  z  ->  Pred ( R ,  A ,  X )  C_  z
) ) )
2120adantr 452 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)  ->  ( ( X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_  dom  g )  <->  ( X  e.  z  ->  Pred ( R ,  A ,  X )  C_  z
) ) )
2216, 21mpbird 224 . . . . . . . . 9  |-  ( ( g  Fn  z  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)  ->  ( X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_  dom  g ) )
2322adantrl 697 . . . . . . . 8  |-  ( ( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z ) )  -> 
( X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_ 
dom  g ) )
24233adant3 977 . . . . . . 7  |-  ( ( g  Fn  z  /\  ( z  C_  A  /\  A. w  e.  z 
Pred ( R ,  A ,  w )  C_  z )  /\  A. w  e.  z  (
g `  w )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w ) ) ) )  ->  ( X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_  dom  g ) )
2524exlimiv 1644 . . . . . 6  |-  ( E. z ( g  Fn  z  /\  ( z 
C_  A  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  Pred ( R ,  A ,  w )
) ) )  -> 
( X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_ 
dom  g ) )
2612, 25sylbi 188 . . . . 5  |-  ( g  e.  { 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 )
) ) ) }  ->  ( X  e. 
dom  g  ->  Pred ( R ,  A ,  X )  C_  dom  g ) )
2726reximia 2803 . . . 4  |-  ( E. g  e.  { 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 )
) ) ) } X  e.  dom  g  ->  E. g  e.  {
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 ) ) ) ) } Pred ( R ,  A ,  X )  C_  dom  g )
28 ssiun 4125 . . . 4  |-  ( E. g  e.  { 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 )
) ) ) }
Pred ( R ,  A ,  X )  C_ 
dom  g  ->  Pred ( R ,  A ,  X )  C_  U_ g  e.  { 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 )
) ) ) } dom  g )
2927, 28syl 16 . . 3  |-  ( E. g  e.  { 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 )
) ) ) } X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_ 
U_ g  e.  {
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 ) ) ) ) } dom  g
)
309, 29sylbi 188 . 2  |-  ( X  e.  dom  F  ->  Pred ( R ,  A ,  X )  C_  U_ g  e.  { 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 )
) ) ) } dom  g )
3130, 6syl6sseqr 3387 1  |-  ( X  e.  dom  F  ->  Pred ( R ,  A ,  X )  C_  dom  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698    C_ wss 3312   U.cuni 4007   U_ciun 4085   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446   Predcpred 25430  wrecscwrecs 25522
This theorem is referenced by:  wfrlem10  25539  wfrlem14  25543  wfrlem15  25544
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-pred 25431  df-wrecs 25523
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