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Theorem wrecseq123 27721
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 3381 . . . . . . . 8  |-  ( A  =  B  ->  (
x  C_  A  <->  x  C_  B
) )
213ad2ant2 1010 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( x  C_  A  <->  x 
C_  B ) )
3 predeq1 27630 . . . . . . . . . . 11  |-  ( R  =  S  ->  Pred ( R ,  A , 
y )  =  Pred ( S ,  A , 
y ) )
4 predeq2 27631 . . . . . . . . . . 11  |-  ( A  =  B  ->  Pred ( S ,  A , 
y )  =  Pred ( S ,  B , 
y ) )
53, 4sylan9eq 2495 . . . . . . . . . 10  |-  ( ( R  =  S  /\  A  =  B )  ->  Pred ( R ,  A ,  y )  =  Pred ( S ,  B ,  y )
)
653adant3 1008 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  Pred ( R ,  A ,  y )  =  Pred ( S ,  B ,  y )
)
76sseq1d 3386 . . . . . . . 8  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( Pred ( R ,  A ,  y )  C_  x  <->  Pred ( S ,  B ,  y )  C_  x )
)
87ralbidv 2738 . . . . . . 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 710 . . . . . 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 990 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  F  =  G )
115reseq2d 5113 . . . . . . . . . 10  |-  ( ( R  =  S  /\  A  =  B )  ->  ( f  |`  Pred ( R ,  A , 
y ) )  =  ( f  |`  Pred ( S ,  B , 
y ) ) )
12113adant3 1008 . . . . . . . . 9  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( f  |`  Pred ( R ,  A , 
y ) )  =  ( f  |`  Pred ( S ,  B , 
y ) ) )
1310, 12fveq12d 5700 . . . . . . . 8  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( F `  (
f  |`  Pred ( R ,  A ,  y )
) )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) )
1413eqeq2d 2454 . . . . . . 7  |-  ( ( R  =  S  /\  A  =  B  /\  F  =  G )  ->  ( ( f `  y )  =  ( F `  ( f  |`  Pred ( R ,  A ,  y )
) )  <->  ( f `  y )  =  ( G `  ( f  |`  Pred ( S ,  B ,  y )
) ) ) )
1514ralbidv 2738 . . . . . 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 1292 . . . . 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 1680 . . . 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 2560 . . 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 4104 . 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 27720 . 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 27720 . 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 2500 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 369    /\ w3a 965    = wceq 1369   E.wex 1586   {cab 2429   A.wral 2718    C_ wss 3331   U.cuni 4094    |` cres 4845    Fn wfn 5416   ` cfv 5421   Predcpred 27627  wrecscwrecs 27719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-xp 4849  df-cnv 4851  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fv 5429  df-pred 27628  df-wrecs 27720
This theorem is referenced by:  wrecseq1  27723  wrecseq2  27724  wrecseq3  27725
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