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Theorem frrlem5e 29000
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 5212 . . . 4  |-  dom  U. C  =  U_ z  e.  C  dom  z
21eleq2i 2545 . . 3  |-  ( X  e.  dom  U. C  <->  X  e.  U_ z  e.  C  dom  z )
3 eliun 4330 . . 3  |-  ( X  e.  U_ z  e.  C  dom  z  <->  E. z  e.  C  X  e.  dom  z )
42, 3bitri 249 . 2  |-  ( X  e.  dom  U. C  <->  E. z  e.  C  X  e.  dom  z )
5 ssel2 3499 . . . . 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 28992 . . . . . . 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 2594 . . . . . 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 5680 . . . . . . . . 9  |-  ( z  Fn  w  ->  dom  z  =  w )
10 predeq3 28853 . . . . . . . . . . . . 13  |-  ( t  =  X  ->  Pred ( R ,  A , 
t )  =  Pred ( R ,  A ,  X ) )
1110sseq1d 3531 . . . . . . . . . . . 12  |-  ( t  =  X  ->  ( Pred ( R ,  A ,  t )  C_  w 
<-> 
Pred ( R ,  A ,  X )  C_  w ) )
1211rspccv 3211 . . . . . . . . . . 11  |-  ( A. t  e.  w  Pred ( R ,  A , 
t )  C_  w  ->  ( X  e.  w  ->  Pred ( R ,  A ,  X )  C_  w ) )
13123ad2ant2 1018 . . . . . . . . . 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 2540 . . . . . . . . . . 11  |-  ( dom  z  =  w  -> 
( X  e.  dom  z 
<->  X  e.  w ) )
15 sseq2 3526 . . . . . . . . . . 11  |-  ( dom  z  =  w  -> 
( Pred ( R ,  A ,  X )  C_ 
dom  z  <->  Pred ( R ,  A ,  X
)  C_  w )
)
1614, 15imbi12d 320 . . . . . . . . . 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 221 . . . . . . . . 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 16 . . . . . . . 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 429 . . . . . . 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 1698 . . . . . 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 195 . . . . 5  |-  ( z  e.  B  ->  ( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_  dom  z ) )
225, 21syl 16 . . . 4  |-  ( ( C  C_  B  /\  z  e.  C )  ->  ( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  z ) )
23 dmeq 5203 . . . . . . . . . 10  |-  ( w  =  z  ->  dom  w  =  dom  z )
2423sseq2d 3532 . . . . . . . . 9  |-  ( w  =  z  ->  ( Pred ( R ,  A ,  X )  C_  dom  w 
<-> 
Pred ( R ,  A ,  X )  C_ 
dom  z ) )
2524rspcev 3214 . . . . . . . 8  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  E. w  e.  C  Pred ( R ,  A ,  X
)  C_  dom  w )
26 ssiun 4367 . . . . . . . 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 16 . . . . . . 7  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  Pred ( R ,  A ,  X )  C_  U_ w  e.  C  dom  w )
28 dmuni 5212 . . . . . . 7  |-  dom  U. C  =  U_ w  e.  C  dom  w
2927, 28syl6sseqr 3551 . . . . . 6  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  Pred ( R ,  A ,  X )  C_  dom  U. C )
3029ex 434 . . . . 5  |-  ( z  e.  C  ->  ( Pred ( R ,  A ,  X )  C_  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  U. C ) )
3130adantl 466 . . . 4  |-  ( ( C  C_  B  /\  z  e.  C )  ->  ( Pred ( R ,  A ,  X
)  C_  dom  z  ->  Pred ( R ,  A ,  X )  C_  dom  U. C ) )
3222, 31syld 44 . . 3  |-  ( ( C  C_  B  /\  z  e.  C )  ->  ( X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_ 
dom  U. C ) )
3332rexlimdva 2955 . 2  |-  ( C 
C_  B  ->  ( E. z  e.  C  X  e.  dom  z  ->  Pred ( R ,  A ,  X )  C_  dom  U. C ) )
344, 33syl5bi 217 1  |-  ( C 
C_  B  ->  ( X  e.  dom  U. C  ->  Pred ( R ,  A ,  X )  C_ 
dom  U. C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767   {cab 2452   A.wral 2814   E.wrex 2815    C_ wss 3476   U.cuni 4245   U_ciun 4325    Fr wfr 4835   Se wse 4836   dom cdm 4999    |` cres 5001    Fn wfn 5583   ` cfv 5588  (class class class)co 6284   Predcpred 28848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596  df-ov 6287  df-pred 28849
This theorem is referenced by: (None)
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