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Theorem wrecseq123 7014
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 3464 . . . . . . . 8  |-  ( A  =  B  ->  (
x  C_  A  <->  x  C_  B
) )
213ad2ant2 1019 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( x  C_  A  <->  x 
C_  B ) )
3 predeq1 5369 . . . . . . . . . . 11  |-  ( R  =  S  ->  Pred ( R ,  A , 
y )  =  Pred ( S ,  A , 
y ) )
4 predeq2 5370 . . . . . . . . . . 11  |-  ( A  =  B  ->  Pred ( S ,  A , 
y )  =  Pred ( S ,  B , 
y ) )
53, 4sylan9eq 2463 . . . . . . . . . 10  |-  ( ( R  =  S  /\  A  =  B )  ->  Pred ( R ,  A ,  y )  =  Pred ( S ,  B ,  y )
)
653adant3 1017 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  Pred ( R ,  A ,  y )  =  Pred ( S ,  B ,  y )
)
76sseq1d 3469 . . . . . . . 8  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( Pred ( R ,  A ,  y )  C_  x  <->  Pred ( S ,  B ,  y )  C_  x )
)
87ralbidv 2843 . . . . . . 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 709 . . . . . 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 999 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  F  =  G )
115reseq2d 5094 . . . . . . . . . 10  |-  ( ( R  =  S  /\  A  =  B )  ->  ( f  |`  Pred ( R ,  A , 
y ) )  =  ( f  |`  Pred ( S ,  B , 
y ) ) )
12113adant3 1017 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( f  |`  Pred ( R ,  A , 
y ) )  =  ( f  |`  Pred ( S ,  B , 
y ) ) )
1310, 12fveq12d 5855 . . . . . . . 8  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( F `  (
f  |`  Pred ( R ,  A ,  y )
) )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) )
1413eqeq2d 2416 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( ( f `  y )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) )  <->  ( f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) ) )
1514ralbidv 2843 . . . . . 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 1304 . . . . 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 1735 . . . 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 2538 . . 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 4201 . 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 7013 . 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 7013 . 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 2468 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 184    /\ wa 367    /\ w3a 974    = wceq 1405   E.wex 1633   {cab 2387   A.wral 2754    C_ wss 3414   U.cuni 4191    |` cres 4825   Predcpred 5366    Fn wfn 5564   ` cfv 5569  wrecscwrecs 7012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-iota 5533  df-fv 5577  df-wrecs 7013
This theorem is referenced by:  wrecseq1  7016  wrecseq2  7017  wrecseq3  7018
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