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Theorem frrlem6 30310
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 4976 . . . 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 30302 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
5 funrel 5618 . . . . 5  |-  ( Fun  g  ->  Rel  g )
64, 5syl 17 . . . 4  |-  ( g  e.  B  ->  Rel  g )
72, 6mprgbir 2796 . . 3  |-  Rel  U. B
8 releq 4937 . . 3  |-  ( F  =  U. B  -> 
( Rel  F  <->  Rel  U. B
) )
97, 8mpbiri 236 . 2  |-  ( F  =  U. B  ->  Rel  F )
101, 9ax-mp 5 1  |-  Rel  F
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1870   {cab 2414   A.wral 2782    C_ wss 3442   U.cuni 4222    |` cres 4856   Rel wrel 4859   Predcpred 5398   Fun wfun 5595    Fn wfn 5596   ` cfv 5601  (class class class)co 6305
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-iun 4304  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ov 6308
This theorem is referenced by: (None)
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