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Theorem wrecseq123 7034
Description: General equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)
Assertion
Ref Expression
wrecseq123  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  -> wrecs ( R ,  A ,  F )  = wrecs ( S ,  B ,  G ) )

Proof of Theorem wrecseq123
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq2 3486 . . . . . . . 8  |-  ( A  =  B  ->  (
x  C_  A  <->  x  C_  B
) )
213ad2ant2 1027 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( x  C_  A  <->  x 
C_  B ) )
3 predeq1 5398 . . . . . . . . . . 11  |-  ( R  =  S  ->  Pred ( R ,  A , 
y )  =  Pred ( S ,  A , 
y ) )
4 predeq2 5399 . . . . . . . . . . 11  |-  ( A  =  B  ->  Pred ( S ,  A , 
y )  =  Pred ( S ,  B , 
y ) )
53, 4sylan9eq 2483 . . . . . . . . . 10  |-  ( ( R  =  S  /\  A  =  B )  ->  Pred ( R ,  A ,  y )  =  Pred ( S ,  B ,  y )
)
653adant3 1025 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  Pred ( R ,  A ,  y )  =  Pred ( S ,  B ,  y )
)
76sseq1d 3491 . . . . . . . 8  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( Pred ( R ,  A ,  y )  C_  x  <->  Pred ( S ,  B ,  y )  C_  x )
)
87ralbidv 2864 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( A. y  e.  x  Pred ( R ,  A ,  y )  C_  x  <->  A. y  e.  x  Pred ( S ,  B ,  y )  C_  x ) )
92, 8anbi12d 715 . . . . . 6  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A ,  y )  C_  x )  <->  ( x  C_  B  /\  A. y  e.  x  Pred ( S ,  B ,  y )  C_  x )
) )
10 simp3 1007 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  F  =  G )
115reseq2d 5121 . . . . . . . . . 10  |-  ( ( R  =  S  /\  A  =  B )  ->  ( f  |`  Pred ( R ,  A , 
y ) )  =  ( f  |`  Pred ( S ,  B , 
y ) ) )
12113adant3 1025 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( f  |`  Pred ( R ,  A , 
y ) )  =  ( f  |`  Pred ( S ,  B , 
y ) ) )
1310, 12fveq12d 5884 . . . . . . . 8  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( F `  (
f  |`  Pred ( R ,  A ,  y )
) )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) )
1413eqeq2d 2436 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( ( f `  y )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) )  <->  ( f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) ) )
1514ralbidv 2864 . . . . . 6  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) )  <->  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) ) )
169, 153anbi23d 1338 . . . . 5  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( ( f  Fn  x  /\  ( x 
C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x
)  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) ) )  <->  ( f  Fn  x  /\  (
x  C_  B  /\  A. y  e.  x  Pred ( S ,  B , 
y )  C_  x
)  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) ) ) )
1716exbidv 1758 . . . 4  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( E. x ( f  Fn  x  /\  ( x  C_  A  /\  A. y  e.  x  Pred ( R ,  A , 
y )  C_  x
)  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) ) )  <->  E. x
( f  Fn  x  /\  ( x  C_  B  /\  A. y  e.  x  Pred ( S ,  B ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B , 
y ) ) ) ) ) )
1817abbidv 2558 . . 3  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  { 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 )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }  =  { f  |  E. x ( f  Fn  x  /\  (
x  C_  B  /\  A. y  e.  x  Pred ( S ,  B , 
y )  C_  x
)  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) ) } )
1918unieqd 4226 . 2  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  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 )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) ) ) }  =  U. { f  |  E. x ( f  Fn  x  /\  ( x  C_  B  /\  A. y  e.  x  Pred ( S ,  B , 
y )  C_  x
)  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) ) } )
20 df-wrecs 7033 . 2  |- wrecs ( R ,  A ,  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 )  =  ( F `  ( f  |`  Pred ( R ,  A , 
y ) ) ) ) }
21 df-wrecs 7033 . 2  |- wrecs ( S ,  B ,  G
)  =  U. {
f  |  E. x
( f  Fn  x  /\  ( x  C_  B  /\  A. y  e.  x  Pred ( S ,  B ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B , 
y ) ) ) ) }
2219, 20, 213eqtr4g 2488 1  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  -> wrecs ( R ,  A ,  F )  = wrecs ( S ,  B ,  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659   {cab 2407   A.wral 2775    C_ wss 3436   U.cuni 4216    |` cres 4852   Predcpred 5395    Fn wfn 5593   ` cfv 5598  wrecscwrecs 7032
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 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
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 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-xp 4856  df-cnv 4858  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-iota 5562  df-fv 5606  df-wrecs 7033
This theorem is referenced by:  wrecseq1  7036  wrecseq2  7037  wrecseq3  7038
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