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Theorem frrlem5e 24360
Description: Lemma for founded recursion. The domain of the union of a subset of  B is closed under predecessors. (Contributed by Paul Chapman, 1-May-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
frrlem5e  |-  ( C 
C_  B  ->  ( X  e.  dom  U. C  ->  Pred ( R ,  A ,  X )  C_ 
dom  U. C ) )
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)    X( x, y, f)

Proof of Theorem frrlem5e
Dummy variables  z 
t  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmuni 4904 . . . 4  |-  dom  U. C  =  U_ z  e.  C  dom  z
21eleq2i 2360 . . 3  |-  ( X  e.  dom  U. C  <->  X  e.  U_ z  e.  C  dom  z )
3 eliun 3925 . . 3  |-  ( X  e.  U_ z  e.  C  dom  z  <->  E. z  e.  C  X  e.  dom  z )
42, 3bitri 240 . 2  |-  ( X  e.  dom  U. C  <->  E. z  e.  C  X  e.  dom  z )
5 ssel2 3188 . . . . 5  |-  ( ( C  C_  B  /\  z  e.  C )  ->  z  e.  B )
6 frrlem5.3 . . . . . . . 8  |-  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 ) ) ) ) ) }
76frrlem1 24352 . . . . . . 7  |-  B  =  { z  |  E. w ( z  Fn  w  /\  ( w 
C_  A  /\  A. t  e.  w  Pred ( R ,  A , 
t )  C_  w  /\  A. t  e.  w  ( z `  t
)  =  ( t G ( z  |`  Pred ( R ,  A ,  t ) ) ) ) ) }
87abeq2i 2403 . . . . . 6  |-  ( z  e.  B  <->  E. w
( z  Fn  w  /\  ( w  C_  A  /\  A. t  e.  w  Pred ( R ,  A ,  t )  C_  w  /\  A. t  e.  w  ( z `  t )  =  ( t G ( z  |`  Pred ( R ,  A ,  t )
) ) ) ) )
9 fndm 5359 . . . . . . . . 9  |-  ( z  Fn  w  ->  dom  z  =  w )
10 predeq3 24242 . . . . . . . . . . . . 13  |-  ( t  =  X  ->  Pred ( R ,  A , 
t )  =  Pred ( R ,  A ,  X ) )
1110sseq1d 3218 . . . . . . . . . . . 12  |-  ( t  =  X  ->  ( Pred ( R ,  A ,  t )  C_  w 
<-> 
Pred ( R ,  A ,  X )  C_  w ) )
1211rspccv 2894 . . . . . . . . . . 11  |-  ( A. t  e.  w  Pred ( R ,  A , 
t )  C_  w  ->  ( X  e.  w  ->  Pred ( R ,  A ,  X )  C_  w ) )
13123ad2ant2 977 . . . . . . . . . 10  |-  ( ( w  C_  A  /\  A. t  e.  w  Pred ( R ,  A , 
t )  C_  w  /\  A. t  e.  w  ( z `  t
)  =  ( t G ( z  |`  Pred ( R ,  A ,  t ) ) ) )  ->  ( X  e.  w  ->  Pred ( R ,  A ,  X )  C_  w
) )
14 eleq2 2357 . . . . . . . . . . 11  |-  ( dom  z  =  w  -> 
( X  e.  dom  z 
<->  X  e.  w ) )
15 sseq2 3213 . . . . . . . . . . 11  |-  ( dom  z  =  w  -> 
( Pred ( R ,  A ,  X )  C_ 
dom  z  <->  Pred ( R ,  A ,  X
)  C_  w )
)
1614, 15imbi12d 311 . . . . . . . . . 10  |-  ( dom  z  =  w  -> 
( ( X  e. 
dom  z  ->  Pred ( R ,  A ,  X )  C_  dom  z )  <->  ( X  e.  w  ->  Pred ( R ,  A ,  X )  C_  w
) ) )
1713, 16syl5ibr 212 . . . . . . . . 9  |-  ( dom  z  =  w  -> 
( ( w  C_  A  /\  A. t  e.  w  Pred ( R ,  A ,  t )  C_  w  /\  A. t  e.  w  ( z `  t )  =  ( t G ( z  |`  Pred ( R ,  A ,  t )
) ) )  -> 
( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  z ) ) )
189, 17syl 15 . . . . . . . 8  |-  ( z  Fn  w  ->  (
( w  C_  A  /\  A. t  e.  w  Pred ( R ,  A ,  t )  C_  w  /\  A. t  e.  w  ( z `  t )  =  ( t G ( z  |`  Pred ( R ,  A ,  t )
) ) )  -> 
( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  z ) ) )
1918imp 418 . . . . . . 7  |-  ( ( z  Fn  w  /\  ( w  C_  A  /\  A. t  e.  w  Pred ( R ,  A , 
t )  C_  w  /\  A. t  e.  w  ( z `  t
)  =  ( t G ( z  |`  Pred ( R ,  A ,  t ) ) ) ) )  -> 
( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  z ) )
2019exlimiv 1624 . . . . . 6  |-  ( E. w ( z  Fn  w  /\  ( w 
C_  A  /\  A. t  e.  w  Pred ( R ,  A , 
t )  C_  w  /\  A. t  e.  w  ( z `  t
)  =  ( t G ( z  |`  Pred ( R ,  A ,  t ) ) ) ) )  -> 
( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  z ) )
218, 20sylbi 187 . . . . 5  |-  ( z  e.  B  ->  ( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_  dom  z ) )
225, 21syl 15 . . . 4  |-  ( ( C  C_  B  /\  z  e.  C )  ->  ( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  z ) )
23 dmeq 4895 . . . . . . . . . 10  |-  ( w  =  z  ->  dom  w  =  dom  z )
2423sseq2d 3219 . . . . . . . . 9  |-  ( w  =  z  ->  ( Pred ( R ,  A ,  X )  C_  dom  w 
<-> 
Pred ( R ,  A ,  X )  C_ 
dom  z ) )
2524rspcev 2897 . . . . . . . 8  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  E. w  e.  C  Pred ( R ,  A ,  X
)  C_  dom  w )
26 ssiun 3960 . . . . . . . 8  |-  ( E. w  e.  C  Pred ( R ,  A ,  X )  C_  dom  w  ->  Pred ( R ,  A ,  X )  C_ 
U_ w  e.  C  dom  w )
2725, 26syl 15 . . . . . . 7  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  Pred ( R ,  A ,  X )  C_  U_ w  e.  C  dom  w )
28 dmuni 4904 . . . . . . 7  |-  dom  U. C  =  U_ w  e.  C  dom  w
2927, 28syl6sseqr 3238 . . . . . 6  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  Pred ( R ,  A ,  X )  C_  dom  U. C )
3029ex 423 . . . . 5  |-  ( z  e.  C  ->  ( Pred ( R ,  A ,  X )  C_  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  U. C ) )
3130adantl 452 . . . 4  |-  ( ( C  C_  B  /\  z  e.  C )  ->  ( Pred ( R ,  A ,  X
)  C_  dom  z  ->  Pred ( R ,  A ,  X )  C_  dom  U. C ) )
3222, 31syld 40 . . 3  |-  ( ( C  C_  B  /\  z  e.  C )  ->  ( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  U. C ) )
3332rexlimdva 2680 . 2  |-  ( C 
C_  B  ->  ( E. z  e.  C  X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_  dom  U. C ) )
344, 33syl5bi 208 1  |-  ( C 
C_  B  ->  ( X  e.  dom  U. C  ->  Pred ( R ,  A ,  X )  C_ 
dom  U. C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557    C_ wss 3165   U.cuni 3843   U_ciun 3921    Fr wfr 4365   Se wse 4366   dom cdm 4705    |` cres 4707    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   Predcpred 24238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5877  df-pred 24239
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