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Theorem frrlem5d 27912
Description: Lemma for founded recursion. The domain of the union of a subset of  B is a subset of  A. (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
frrlem5d  |-  ( C 
C_  B  ->  dom  U. C  C_  A )
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 frrlem5d
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 dmuni 5150 . 2  |-  dom  U. C  =  U_ g  e.  C  dom  g
2 ssel 3451 . . . . 5  |-  ( C 
C_  B  ->  (
g  e.  C  -> 
g  e.  B ) )
3 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 ) ) ) ) ) }
43frrlem3 27907 . . . . 5  |-  ( g  e.  B  ->  dom  g  C_  A )
52, 4syl6 33 . . . 4  |-  ( C 
C_  B  ->  (
g  e.  C  ->  dom  g  C_  A ) )
65ralrimiv 2823 . . 3  |-  ( C 
C_  B  ->  A. g  e.  C  dom  g  C_  A )
7 iunss 4312 . . 3  |-  ( U_ g  e.  C  dom  g  C_  A  <->  A. g  e.  C  dom  g  C_  A )
86, 7sylibr 212 . 2  |-  ( C 
C_  B  ->  U_ g  e.  C  dom  g  C_  A )
91, 8syl5eqss 3501 1  |-  ( C 
C_  B  ->  dom  U. C  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2436   A.wral 2795    C_ wss 3429   U.cuni 4192   U_ciun 4272    Fr wfr 4777   Se wse 4778   dom cdm 4941    |` cres 4943    Fn wfn 5514   ` cfv 5519  (class class class)co 6193   Predcpred 27761
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 1952  ax-ext 2430
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 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527  df-ov 6196  df-pred 27762
This theorem is referenced by: (None)
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