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Theorem frrlem5b 29557
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 3411 . . . 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 29553 . . . . 5  |-  ( z  e.  B  ->  Fun  z )
4 funrel 5513 . . . . 5  |-  ( Fun  z  ->  Rel  z )
53, 4syl 16 . . . 4  |-  ( z  e.  B  ->  Rel  z )
61, 5syl6 33 . . 3  |-  ( C 
C_  B  ->  (
z  e.  C  ->  Rel  z ) )
76ralrimiv 2794 . 2  |-  ( C 
C_  B  ->  A. z  e.  C  Rel  z )
8 reluni 5037 . 2  |-  ( Rel  U. C  <->  A. z  e.  C  Rel  z )
97, 8sylibr 212 1  |-  ( C 
C_  B  ->  Rel  U. C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826   {cab 2367   A.wral 2732    C_ wss 3389   U.cuni 4163    Fr wfr 4749   Se wse 4750    |` cres 4915   Rel wrel 4918   Fun wfun 5490    Fn wfn 5491   ` cfv 5496  (class class class)co 6196   Predcpred 29408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504  df-ov 6199  df-pred 29409
This theorem is referenced by:  frrlem5c  29558
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