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Theorem frrlem6 23701
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 4808 . . . 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 23693 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
5 funrel 5272 . . . . 5  |-  ( Fun  g  ->  Rel  g )
64, 5syl 15 . . . 4  |-  ( g  e.  B  ->  Rel  g )
72, 6mprgbir 2613 . . 3  |-  Rel  U. B
8 releq 4771 . . 3  |-  ( F  =  U. B  -> 
( Rel  F  <->  Rel  U. B
) )
97, 8mpbiri 224 . 2  |-  ( F  =  U. B  ->  Rel  F )
101, 9ax-mp 8 1  |-  Rel  F
Colors of variables: wff set class
Syntax hints:    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    C_ wss 3152   U.cuni 3827    |` cres 4691   Rel wrel 4694   Fun wfun 5249    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Predcpred 23578
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-pred 23579
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