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Theorem frrlem5d 27726
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 5044 . 2  |-  dom  U. C  =  U_ g  e.  C  dom  g
2 ssel 3345 . . . . 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 27721 . . . . 5  |-  ( g  e.  B  ->  dom  g  C_  A )
52, 4syl6 33 . . . 4  |-  ( C 
C_  B  ->  (
g  e.  C  ->  dom  g  C_  A ) )
65ralrimiv 2793 . . 3  |-  ( C 
C_  B  ->  A. g  e.  C  dom  g  C_  A )
7 iunss 4206 . . 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 3395 1  |-  ( C 
C_  B  ->  dom  U. C  C_  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2424   A.wral 2710    C_ wss 3323   U.cuni 4086   U_ciun 4166    Fr wfr 4671   Se wse 4672   dom cdm 4835    |` cres 4837    Fn wfn 5408   ` cfv 5413  (class class class)co 6086   Predcpred 27575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-ov 6089  df-pred 27576
This theorem is referenced by: (None)
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