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Theorem nfwrecs 7038
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 7036 . 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 1754 . . . . . 6  |-  F/ x  f  Fn  y
3 nfcv 2591 . . . . . . . 8  |-  F/_ x
y
4 nfwrecs.2 . . . . . . . 8  |-  F/_ x A
53, 4nfss 3463 . . . . . . 7  |-  F/ x  y  C_  A
6 nfwrecs.1 . . . . . . . . . 10  |-  F/_ x R
7 nfcv 2591 . . . . . . . . . 10  |-  F/_ x
z
86, 4, 7nfpred 5404 . . . . . . . . 9  |-  F/_ x Pred ( R ,  A ,  z )
98, 3nfss 3463 . . . . . . . 8  |-  F/ x Pred ( R ,  A ,  z )  C_  y
103, 9nfral 2818 . . . . . . 7  |-  F/ x A. z  e.  y  Pred ( R ,  A ,  z )  C_  y
115, 10nfan 1986 . . . . . 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 2591 . . . . . . . . . 10  |-  F/_ x
f
1413, 8nfres 5127 . . . . . . . . 9  |-  F/_ x
( f  |`  Pred ( R ,  A , 
z ) )
1512, 14nffv 5888 . . . . . . . 8  |-  F/_ x
( F `  (
f  |`  Pred ( R ,  A ,  z )
) )
1615nfeq2 2608 . . . . . . 7  |-  F/ x
( f `  z
)  =  ( F `
 ( f  |`  Pred ( R ,  A ,  z ) ) )
173, 16nfral 2818 . . . . . 6  |-  F/ x A. z  e.  y 
( f `  z
)  =  ( F `
 ( f  |`  Pred ( R ,  A ,  z ) ) )
182, 11, 17nf3an 1988 . . . . 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 2006 . . . 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 2595 . . 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 4228 . 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 2589 1  |-  F/_ xwrecs ( R ,  A ,  F )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659   {cab 2414   F/_wnfc 2577   A.wral 2782    C_ wss 3442   U.cuni 4222    |` cres 4856   Predcpred 5398    Fn wfn 5596   ` cfv 5601  wrecscwrecs 7035
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 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
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 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-xp 4860  df-cnv 4862  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-iota 5565  df-fv 5609  df-wrecs 7036
This theorem is referenced by:  nfrecs  7101
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