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Theorem frrlem5d 24288
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 4888 . 2  |-  dom  U. C  =  U_ g  e.  C  dom  g
2 ssel 3174 . . . . 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 24283 . . . . 5  |-  ( g  e.  B  ->  dom  g  C_  A )
52, 4syl6 29 . . . 4  |-  ( C 
C_  B  ->  (
g  e.  C  ->  dom  g  C_  A ) )
65ralrimiv 2625 . . 3  |-  ( C 
C_  B  ->  A. g  e.  C  dom  g  C_  A )
7 iunss 3943 . . 3  |-  ( U_ g  e.  C  dom  g  C_  A  <->  A. g  e.  C  dom  g  C_  A )
86, 7sylibr 203 . 2  |-  ( C 
C_  B  ->  U_ g  e.  C  dom  g  C_  A )
91, 8syl5eqss 3222 1  |-  ( C 
C_  B  ->  dom  U. C  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543    C_ wss 3152   U.cuni 3827   U_ciun 3905    Fr wfr 4349   Se wse 4350   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   Predcpred 24167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5861  df-pred 24168
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