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Theorem nfrecs 6286
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 6284 . 2  |- recs ( F )  =  U. {
a  |  E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c ) ) ) }
2 nfcv 2312 . . . . 5  |-  F/_ x On
3 nfv 1521 . . . . . 6  |-  F/ x  a  Fn  b
4 nfcv 2312 . . . . . . 7  |-  F/_ x
b
5 nfrecs.f . . . . . . . . 9  |-  F/_ x F
6 nfcv 2312 . . . . . . . . 9  |-  F/_ x
( a  |`  c
)
75, 6nffv 5506 . . . . . . . 8  |-  F/_ x
( F `  (
a  |`  c ) )
87nfeq2 2324 . . . . . . 7  |-  F/ x
( a `  c
)  =  ( F `
 ( a  |`  c ) )
94, 8nfralxy 2508 . . . . . 6  |-  F/ x A. c  e.  b 
( a `  c
)  =  ( F `
 ( a  |`  c ) )
103, 9nfan 1558 . . . . 5  |-  F/ x
( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
112, 10nfrexxy 2509 . . . 4  |-  F/ x E. b  e.  On  ( a  Fn  b  /\  A. c  e.  b  ( a `  c
)  =  ( F `
 ( a  |`  c ) ) )
1211nfab 2317 . . 3  |-  F/_ x { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
1312nfuni 3802 . 2  |-  F/_ x U. { a  |  E. b  e.  On  (
a  Fn  b  /\  A. c  e.  b  ( a `  c )  =  ( F `  ( a  |`  c
) ) ) }
141, 13nfcxfr 2309 1  |-  F/_ xrecs ( F )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1348   {cab 2156   F/_wnfc 2299   A.wral 2448   E.wrex 2449   U.cuni 3796   Oncon0 4348    |` cres 4613    Fn wfn 5193   ` cfv 5198  recscrecs 6283
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-recs 6284
This theorem is referenced by:  nffrec  6375
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