Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frrlem6 Unicode version

Theorem frrlem6 25316
Description: Lemma for founded recursion. The union of all acceptable functions is a relationship. (Contributed by Paul Chapman, 21-Apr-2012.)
Hypotheses
Ref Expression
frrlem6.1  |-  B  =  { 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
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
frrlem6.2  |-  F  = 
U. B
Assertion
Ref Expression
frrlem6  |-  Rel  F
Distinct variable groups:    A, f, x, y    f, G, x, y    R, f, x, y
Allowed substitution hints:    B( x, y, f)    F( x, y, f)

Proof of Theorem frrlem6
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 frrlem6.2 . 2  |-  F  = 
U. B
2 reluni 4939 . . . 4  |-  ( Rel  U. B  <->  A. g  e.  B  Rel  g )
3 frrlem6.1 . . . . . 6  |-  B  =  { 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
)  =  ( y G ( f  |`  Pred ( R ,  A ,  y ) ) ) ) ) }
43frrlem2 25308 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
5 funrel 5413 . . . . 5  |-  ( Fun  g  ->  Rel  g )
64, 5syl 16 . . . 4  |-  ( g  e.  B  ->  Rel  g )
72, 6mprgbir 2721 . . 3  |-  Rel  U. B
8 releq 4901 . . 3  |-  ( F  =  U. B  -> 
( Rel  F  <->  Rel  U. B
) )
97, 8mpbiri 225 . 2  |-  ( F  =  U. B  ->  Rel  F )
101, 9ax-mp 8 1  |-  Rel  F
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2375   A.wral 2651    C_ wss 3265   U.cuni 3959    |` cres 4822   Rel wrel 4825   Fun wfun 5390    Fn wfn 5391   ` cfv 5396  (class class class)co 6022   Predcpred 25193
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-ov 6025  df-pred 25194
  Copyright terms: Public domain W3C validator