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Theorem frrlem5b 25592
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 3344 . . . 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 25588 . . . . 5  |-  ( z  e.  B  ->  Fun  z )
4 funrel 5474 . . . . 5  |-  ( Fun  z  ->  Rel  z )
53, 4syl 16 . . . 4  |-  ( z  e.  B  ->  Rel  z )
61, 5syl6 32 . . 3  |-  ( C 
C_  B  ->  (
z  e.  C  ->  Rel  z ) )
76ralrimiv 2790 . 2  |-  ( C 
C_  B  ->  A. z  e.  C  Rel  z )
8 reluni 5000 . 2  |-  ( Rel  U. C  <->  A. z  e.  C  Rel  z )
97, 8sylibr 205 1  |-  ( C 
C_  B  ->  Rel  U. C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707    C_ wss 3322   U.cuni 4017    Fr wfr 4541   Se wse 4542    |` cres 4883   Rel wrel 4886   Fun wfun 5451    Fn wfn 5452   ` cfv 5457  (class class class)co 6084   Predcpred 25443
This theorem is referenced by:  frrlem5c  25593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-ov 6087  df-pred 25444
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