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Theorem frrlem5b 27918
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 3459 . . . 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 27914 . . . . 5  |-  ( z  e.  B  ->  Fun  z )
4 funrel 5544 . . . . 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 2828 . 2  |-  ( C 
C_  B  ->  A. z  e.  C  Rel  z )
8 reluni 5071 . 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 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2439   A.wral 2799    C_ wss 3437   U.cuni 4200    Fr wfr 4785   Se wse 4786    |` cres 4951   Rel wrel 4954   Fun wfun 5521    Fn wfn 5522   ` cfv 5527  (class class class)co 6201   Predcpred 27769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-fv 5535  df-ov 6204  df-pred 27770
This theorem is referenced by:  frrlem5c  27919
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