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Theorem wfrdmcl 7044
Description: Given  F  = wrecs ( R ,  A ,  X )  /\  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
wfrdmcl  |-  ( X  e.  dom  F  ->  Pred ( R ,  A ,  X )  C_  dom  F )

Proof of Theorem wfrdmcl
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 7028 . . . . . . . 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 2449 . . . . . . 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 5048 . . . . . 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 5056 . . . . . 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 2449 . . . . 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 2498 . . . 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 4298 . . . 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 252 . . 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 2420 . . . . . . . 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 7035 . . . . . . 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 2547 . . . . . 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 5395 . . . . . . . . . . . . 13  |-  ( w  =  X  ->  Pred ( R ,  A ,  w )  =  Pred ( R ,  A ,  X ) )
1413sseq1d 3488 . . . . . . . . . . . 12  |-  ( w  =  X  ->  ( Pred ( R ,  A ,  w )  C_  z  <->  Pred ( R ,  A ,  X )  C_  z
) )
1514rspccv 3176 . . . . . . . . . . 11  |-  ( A. w  e.  z  Pred ( R ,  A ,  w )  C_  z  ->  ( X  e.  z  ->  Pred ( R ,  A ,  X )  C_  z ) )
1615adantl 467 . . . . . . . . . 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 5685 . . . . . . . . . . . . 13  |-  ( g  Fn  z  ->  dom  g  =  z )
1817eleq2d 2490 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  ( X  e.  dom  g  <->  X  e.  z ) )
1917sseq2d 3489 . . . . . . . . . . . 12  |-  ( g  Fn  z  ->  ( Pred ( R ,  A ,  X )  C_  dom  g 
<-> 
Pred ( R ,  A ,  X )  C_  z ) )
2018, 19imbi12d 321 . . . . . . . . . . 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 466 . . . . . . . . . 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 235 . . . . . . . . 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 720 . . . . . . . 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 1025 . . . . . . 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 1766 . . . . . 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 198 . . . . 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 2889 . . . 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 4335 . . . 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 17 . . 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 198 . 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 3508 1  |-  ( X  e.  dom  F  ->  Pred ( R ,  A ,  X )  C_  dom  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1867   {cab 2405   A.wral 2773   E.wrex 2774    C_ wss 3433   U.cuni 4213   U_ciun 4293   dom cdm 4846    |` cres 4848   Predcpred 5390    Fn wfn 5588   ` cfv 5593  wrecscwrecs 7027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4477  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-rn 4857  df-res 4858  df-ima 4859  df-pred 5391  df-iota 5557  df-fun 5595  df-fn 5596  df-fv 5601  df-wrecs 7028
This theorem is referenced by:  wfrlem10  7045  wfrlem14  7049  wfrlem15  7050
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