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Theorem frrlem6 24361
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 4824 . . . 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 24353 . . . . 5  |-  ( g  e.  B  ->  Fun  g )
5 funrel 5288 . . . . 5  |-  ( Fun  g  ->  Rel  g )
64, 5syl 15 . . . 4  |-  ( g  e.  B  ->  Rel  g )
72, 6mprgbir 2626 . . 3  |-  Rel  U. B
8 releq 4787 . . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556    C_ wss 3165   U.cuni 3843    |` cres 4707   Rel wrel 4710   Fun wfun 5265    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   Predcpred 24238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877  df-pred 24239
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