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Theorem frrlem5e 24289
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 4888 . . . 4  |-  dom  U. C  =  U_ z  e.  C  dom  z
21eleq2i 2347 . . 3  |-  ( X  e.  dom  U. C  <->  X  e.  U_ z  e.  C  dom  z )
3 eliun 3909 . . 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 3175 . . . . 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 24281 . . . . . . 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 2390 . . . . . 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 5343 . . . . . . . . 9  |-  ( z  Fn  w  ->  dom  z  =  w )
10 predeq3 24171 . . . . . . . . . . . . 13  |-  ( t  =  X  ->  Pred ( R ,  A , 
t )  =  Pred ( R ,  A ,  X ) )
1110sseq1d 3205 . . . . . . . . . . . 12  |-  ( t  =  X  ->  ( Pred ( R ,  A ,  t )  C_  w 
<-> 
Pred ( R ,  A ,  X )  C_  w ) )
1211rspccv 2881 . . . . . . . . . . 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 2344 . . . . . . . . . . 11  |-  ( dom  z  =  w  -> 
( X  e.  dom  z 
<->  X  e.  w ) )
15 sseq2 3200 . . . . . . . . . . 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 1666 . . . . . 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 4879 . . . . . . . . . 10  |-  ( w  =  z  ->  dom  w  =  dom  z )
2423sseq2d 3206 . . . . . . . . 9  |-  ( w  =  z  ->  ( Pred ( R ,  A ,  X )  C_  dom  w 
<-> 
Pred ( R ,  A ,  X )  C_ 
dom  z ) )
2524rspcev 2884 . . . . . . . 8  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  E. w  e.  C  Pred ( R ,  A ,  X
)  C_  dom  w )
26 ssiun 3944 . . . . . . . 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 4888 . . . . . . 7  |-  dom  U. C  =  U_ w  e.  C  dom  w
2927, 28syl6sseqr 3225 . . . . . 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 2667 . 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 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    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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