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Theorem frrlem5b 24844
Description: Lemma for founded recursion. The union of a subclass of  B is a relationship. (Contributed by Paul Chapman, 29-Apr-2012.)
Hypotheses
Ref Expression
frrlem5.1  |-  R  Fr  A
frrlem5.2  |-  R Se  A
frrlem5.3  |-  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 ) ) ) ) ) }
Assertion
Ref Expression
frrlem5b  |-  ( C 
C_  B  ->  Rel  U. C )
Distinct variable groups:    A, f, x, y    f, G, x, y    R, f, x, y   
x, B
Allowed substitution hints:    B( y, f)    C( x, y, f)

Proof of Theorem frrlem5b
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ssel 3250 . . . 4  |-  ( C 
C_  B  ->  (
z  e.  C  -> 
z  e.  B ) )
2 frrlem5.3 . . . . . 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 ) ) ) ) ) }
32frrlem2 24840 . . . . 5  |-  ( z  e.  B  ->  Fun  z )
4 funrel 5354 . . . . 5  |-  ( Fun  z  ->  Rel  z )
53, 4syl 15 . . . 4  |-  ( z  e.  B  ->  Rel  z )
61, 5syl6 29 . . 3  |-  ( C 
C_  B  ->  (
z  e.  C  ->  Rel  z ) )
76ralrimiv 2701 . 2  |-  ( C 
C_  B  ->  A. z  e.  C  Rel  z )
8 reluni 4890 . 2  |-  ( Rel  U. C  <->  A. z  e.  C  Rel  z )
97, 8sylibr 203 1  |-  ( C 
C_  B  ->  Rel  U. C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344   A.wral 2619    C_ wss 3228   U.cuni 3908    Fr wfr 4431   Se wse 4432    |` cres 4773   Rel wrel 4776   Fun wfun 5331    Fn wfn 5332   ` cfv 5337  (class class class)co 5945   Predcpred 24725
This theorem is referenced by:  frrlem5c  24845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-fv 5345  df-ov 5948  df-pred 24726
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