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Theorem frrlem6 25583
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 4989 . . . 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 25575 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
5 funrel 5463 . . . . 5  |-  ( Fun  g  ->  Rel  g )
64, 5syl 16 . . . 4  |-  ( g  e.  B  ->  Rel  g )
72, 6mprgbir 2768 . . 3  |-  Rel  U. B
8 releq 4951 . . 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697    C_ wss 3312   U.cuni 4007    |` cres 4872   Rel wrel 4875   Fun wfun 5440    Fn wfn 5441   ` cfv 5446  (class class class)co 6073   Predcpred 25430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ov 6076  df-pred 25431
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