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Theorem nfwrecs 28943
Description: Bound-variable hypothesis builder for the well-founded recursive function generator. (Contributed by Scott Fenton, 9-Jun-2018.)
Hypotheses
Ref Expression
nfwrecs.1  |-  F/_ x R
nfwrecs.2  |-  F/_ x A
nfwrecs.3  |-  F/_ x F
Assertion
Ref Expression
nfwrecs  |-  F/_ xwrecs ( R ,  A ,  F )

Proof of Theorem nfwrecs
Dummy variables  f 
y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wrecs 28941 . 2  |- wrecs ( R ,  A ,  F
)  =  U. {
f  |  E. y
( f  Fn  y  /\  ( y  C_  A  /\  A. z  e.  y 
Pred ( R ,  A ,  z )  C_  y )  /\  A. z  e.  y  (
f `  z )  =  ( F `  ( f  |`  Pred ( R ,  A , 
z ) ) ) ) }
2 nfv 1683 . . . . . 6  |-  F/ x  f  Fn  y
3 nfcv 2629 . . . . . . . 8  |-  F/_ x
y
4 nfwrecs.2 . . . . . . . 8  |-  F/_ x A
53, 4nfss 3497 . . . . . . 7  |-  F/ x  y  C_  A
6 nfwrecs.1 . . . . . . . . . 10  |-  F/_ x R
7 nfcv 2629 . . . . . . . . . 10  |-  F/_ x
z
86, 4, 7nfpred 28854 . . . . . . . . 9  |-  F/_ x Pred ( R ,  A ,  z )
98, 3nfss 3497 . . . . . . . 8  |-  F/ x Pred ( R ,  A ,  z )  C_  y
103, 9nfral 2850 . . . . . . 7  |-  F/ x A. z  e.  y  Pred ( R ,  A ,  z )  C_  y
115, 10nfan 1875 . . . . . 6  |-  F/ x
( y  C_  A  /\  A. z  e.  y 
Pred ( R ,  A ,  z )  C_  y )
12 nfwrecs.3 . . . . . . . . 9  |-  F/_ x F
13 nfcv 2629 . . . . . . . . . 10  |-  F/_ x
f
1413, 8nfres 5275 . . . . . . . . 9  |-  F/_ x
( f  |`  Pred ( R ,  A , 
z ) )
1512, 14nffv 5873 . . . . . . . 8  |-  F/_ x
( F `  (
f  |`  Pred ( R ,  A ,  z )
) )
1615nfeq2 2646 . . . . . . 7  |-  F/ x
( f `  z
)  =  ( F `
 ( f  |`  Pred ( R ,  A ,  z ) ) )
173, 16nfral 2850 . . . . . 6  |-  F/ x A. z  e.  y 
( f `  z
)  =  ( F `
 ( f  |`  Pred ( R ,  A ,  z ) ) )
182, 11, 17nf3an 1877 . . . . 5  |-  F/ x
( f  Fn  y  /\  ( y  C_  A  /\  A. z  e.  y 
Pred ( R ,  A ,  z )  C_  y )  /\  A. z  e.  y  (
f `  z )  =  ( F `  ( f  |`  Pred ( R ,  A , 
z ) ) ) )
1918nfex 1895 . . . 4  |-  F/ x E. y ( f  Fn  y  /\  ( y 
C_  A  /\  A. z  e.  y  Pred ( R ,  A , 
z )  C_  y
)  /\  A. z  e.  y  ( f `  z )  =  ( F `  ( f  |`  Pred ( R ,  A ,  z )
) ) )
2019nfab 2633 . . 3  |-  F/_ x { f  |  E. y ( f  Fn  y  /\  ( y 
C_  A  /\  A. z  e.  y  Pred ( R ,  A , 
z )  C_  y
)  /\  A. z  e.  y  ( f `  z )  =  ( F `  ( f  |`  Pred ( R ,  A ,  z )
) ) ) }
2120nfuni 4251 . 2  |-  F/_ x U. { f  |  E. y ( f  Fn  y  /\  ( y 
C_  A  /\  A. z  e.  y  Pred ( R ,  A , 
z )  C_  y
)  /\  A. z  e.  y  ( f `  z )  =  ( F `  ( f  |`  Pred ( R ,  A ,  z )
) ) ) }
221, 21nfcxfr 2627 1  |-  F/_ xwrecs ( R ,  A ,  F )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596   {cab 2452   F/_wnfc 2615   A.wral 2814    C_ wss 3476   U.cuni 4245    |` cres 5001    Fn wfn 5583   ` cfv 5588   Predcpred 28848  wrecscwrecs 28940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fv 5596  df-pred 28849  df-wrecs 28941
This theorem is referenced by: (None)
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