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Theorem frrlem5d 25589
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 5079 . 2  |-  dom  U. C  =  U_ g  e.  C  dom  g
2 ssel 3342 . . . . 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 25584 . . . . 5  |-  ( g  e.  B  ->  dom  g  C_  A )
52, 4syl6 31 . . . 4  |-  ( C 
C_  B  ->  (
g  e.  C  ->  dom  g  C_  A ) )
65ralrimiv 2788 . . 3  |-  ( C 
C_  B  ->  A. g  e.  C  dom  g  C_  A )
7 iunss 4132 . . 3  |-  ( U_ g  e.  C  dom  g  C_  A  <->  A. g  e.  C  dom  g  C_  A )
86, 7sylibr 204 . 2  |-  ( C 
C_  B  ->  U_ g  e.  C  dom  g  C_  A )
91, 8syl5eqss 3392 1  |-  ( C 
C_  B  ->  dom  U. C  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   A.wral 2705    C_ wss 3320   U.cuni 4015   U_ciun 4093    Fr wfr 4538   Se wse 4539   dom cdm 4878    |` cres 4880    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   Predcpred 25438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462  df-ov 6084  df-pred 25439
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