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Theorem frrlem5d 29362
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 5198 . 2  |-  dom  U. C  =  U_ g  e.  C  dom  g
2 ssel 3480 . . . . 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 29357 . . . . 5  |-  ( g  e.  B  ->  dom  g  C_  A )
52, 4syl6 33 . . . 4  |-  ( C 
C_  B  ->  (
g  e.  C  ->  dom  g  C_  A ) )
65ralrimiv 2853 . . 3  |-  ( C 
C_  B  ->  A. g  e.  C  dom  g  C_  A )
7 iunss 4352 . . 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 3530 1  |-  ( C 
C_  B  ->  dom  U. C  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381   E.wex 1597    e. wcel 1802   {cab 2426   A.wral 2791    C_ wss 3458   U.cuni 4230   U_ciun 4311    Fr wfr 4821   Se wse 4822   dom cdm 4985    |` cres 4987    Fn wfn 5569   ` cfv 5574  (class class class)co 6277   Predcpred 29211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-fv 5582  df-ov 6280  df-pred 29212
This theorem is referenced by: (None)
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