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Theorem wfrlem9 25478
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.)
Hypotheses
Ref Expression
wfrlem6.1  |-  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 )
) ) ) }
wfrlem6.2  |-  F  = 
U. B
Assertion
Ref Expression
wfrlem9  |-  ( X  e.  dom  F  ->  Pred ( R ,  A ,  X )  C_  dom  F )
Distinct variable groups:    A, f, x, y    f, G, x, y    R, f, x, y
Allowed substitution hints:    B( x, y, f)    F( x, y, f)    X( x, y, f)

Proof of Theorem wfrlem9
Dummy variables  g  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wfrlem6.2 . . . . . . 7  |-  F  = 
U. B
21dmeqi 5030 . . . . . 6  |-  dom  F  =  dom  U. B
3 dmuni 5038 . . . . . 6  |-  dom  U. B  =  U_ g  e.  B  dom  g
42, 3eqtri 2424 . . . . 5  |-  dom  F  =  U_ g  e.  B  dom  g
54eleq2i 2468 . . . 4  |-  ( X  e.  dom  F  <->  X  e.  U_ g  e.  B  dom  g )
6 eliun 4057 . . . 4  |-  ( X  e.  U_ g  e.  B  dom  g  <->  E. g  e.  B  X  e.  dom  g )
75, 6bitri 241 . . 3  |-  ( X  e.  dom  F  <->  E. g  e.  B  X  e.  dom  g )
8 wfrlem6.1 . . . . . . . 8  |-  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 )
) ) ) }
98wfrlem1 25470 . . . . . . 7  |-  B  =  { 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 )
) ) ) }
109abeq2i 2511 . . . . . 6  |-  ( g  e.  B  <->  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 ) ) ) ) )
11 predeq3 25385 . . . . . . . . . . . . 13  |-  ( w  =  X  ->  Pred ( R ,  A ,  w )  =  Pred ( R ,  A ,  X ) )
1211sseq1d 3335 . . . . . . . . . . . 12  |-  ( w  =  X  ->  ( Pred ( R ,  A ,  w )  C_  z  <->  Pred ( R ,  A ,  X )  C_  z
) )
1312rspccv 3009 . . . . . . . . . . 11  |-  ( A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  ->  ( X  e.  z  ->  Pred ( R ,  A ,  X )  C_  z ) )
1413adantl 453 . . . . . . . . . 10  |-  ( ( g  Fn  z  /\  A. w  e.  z  Pred ( R ,  A ,  w )  C_  z
)  ->  ( X  e.  z  ->  Pred ( R ,  A ,  X )  C_  z
) )
15 fndm 5503 . . . . . . . . . . . . 13  |-  ( g  Fn  z  ->  dom  g  =  z )
1615eleq2d 2471 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  ( X  e.  dom  g  <->  X  e.  z ) )
1715sseq2d 3336 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  ( Pred ( R ,  A ,  X )  C_  dom  g 
<-> 
Pred ( R ,  A ,  X )  C_  z ) )
1816, 17imbi12d 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
) ) )
1918adantr 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
) ) )
2014, 19mpbird 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 ) )
2120adantrl 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 ) )
22213adant3 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 ) )
2322exlimiv 1641 . . . . . 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 ) )
2410, 23sylbi 188 . . . . 5  |-  ( g  e.  B  ->  ( X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_  dom  g ) )
2524reximia 2771 . . . 4  |-  ( E. g  e.  B  X  e.  dom  g  ->  E. g  e.  B  Pred ( R ,  A ,  X
)  C_  dom  g )
26 ssiun 4093 . . . 4  |-  ( E. g  e.  B  Pred ( R ,  A ,  X )  C_  dom  g  ->  Pred ( R ,  A ,  X )  C_ 
U_ g  e.  B  dom  g )
2725, 26syl 16 . . 3  |-  ( E. g  e.  B  X  e.  dom  g  ->  Pred ( R ,  A ,  X )  C_  U_ g  e.  B  dom  g )
287, 27sylbi 188 . 2  |-  ( X  e.  dom  F  ->  Pred ( R ,  A ,  X )  C_  U_ g  e.  B  dom  g )
2928, 4syl6sseqr 3355 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 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667    C_ wss 3280   U.cuni 3975   U_ciun 4053   dom cdm 4837    |` cres 4839    Fn wfn 5408   ` cfv 5413   Predcpred 25381
This theorem is referenced by:  wfrlem10  25479  wfrlem14  25483  wfrlem15  25484
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-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  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-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-fv 5421  df-pred 25382
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