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Theorem nfrecs 6170
Description: Bound-variable hypothesis builder for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
nfrecs.f  |-  F/_ x F
Assertion
Ref Expression
nfrecs  |-  F/_ xrecs ( F )

Proof of Theorem nfrecs
Dummy variables  a  b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-recs 6168 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
2 nfcv 2256 . . . . 5  |-  F/_ x On
3 nfv 1491 . . . . . 6  |-  F/ x  a  Fn  b
4 nfcv 2256 . . . . . . 7  |-  F/_ x
b
5 nfrecs.f . . . . . . . . 9  |-  F/_ x F
6 nfcv 2256 . . . . . . . . 9  |-  F/_ x
( a  |`  c
)
75, 6nffv 5397 . . . . . . . 8  |-  F/_ x
( F `  (
a  |`  c ) )
87nfeq2 2268 . . . . . . 7  |-  F/ x
( a `  c
)  =  ( F `
 ( a  |`  c ) )
94, 8nfralxy 2446 . . . . . 6  |-  F/ x A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )
103, 9nfan 1527 . . . . 5  |-  F/ x
( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
112, 10nfrexxy 2447 . . . 4  |-  F/ x E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
1211nfab 2261 . . 3  |-  F/_ x { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
1312nfuni 3710 . 2  |-  F/_ x U. { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
141, 13nfcxfr 2253 1  |-  F/_ xrecs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314   {cab 2101   F/_wnfc 2243   A.wral 2391   E.wrex 2392   U.cuni 3704   Oncon0 4253    |` cres 4509    Fn wfn 5086   ` cfv 5091  recscrecs 6167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099  df-recs 6168
This theorem is referenced by:  nffrec  6259
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