Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frrlem5e Structured version   Unicode version

Theorem frrlem5e 27781
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 5054 . . . 4  |-  dom  U. C  =  U_ z  e.  C  dom  z
21eleq2i 2507 . . 3  |-  ( X  e.  dom  U. C  <->  X  e.  U_ z  e.  C  dom  z )
3 eliun 4180 . . 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 3356 . . . . 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 27773 . . . . . . 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 2554 . . . . . 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 5515 . . . . . . . . 9  |-  ( z  Fn  w  ->  dom  z  =  w )
10 predeq3 27634 . . . . . . . . . . . . 13  |-  ( t  =  X  ->  Pred ( R ,  A , 
t )  =  Pred ( R ,  A ,  X ) )
1110sseq1d 3388 . . . . . . . . . . . 12  |-  ( t  =  X  ->  ( Pred ( R ,  A ,  t )  C_  w 
<-> 
Pred ( R ,  A ,  X )  C_  w ) )
1211rspccv 3075 . . . . . . . . . . 11  |-  ( A. t  e.  w  Pred ( R ,  A , 
t )  C_  w  ->  ( X  e.  w  ->  Pred ( R ,  A ,  X )  C_  w ) )
13123ad2ant2 1010 . . . . . . . . . 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 2504 . . . . . . . . . . 11  |-  ( dom  z  =  w  -> 
( X  e.  dom  z 
<->  X  e.  w ) )
15 sseq2 3383 . . . . . . . . . . 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 1688 . . . . . 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 5045 . . . . . . . . . 10  |-  ( w  =  z  ->  dom  w  =  dom  z )
2423sseq2d 3389 . . . . . . . . 9  |-  ( w  =  z  ->  ( Pred ( R ,  A ,  X )  C_  dom  w 
<-> 
Pred ( R ,  A ,  X )  C_ 
dom  z ) )
2524rspcev 3078 . . . . . . . 8  |-  ( ( z  e.  C  /\  Pred ( R ,  A ,  X )  C_  dom  z )  ->  E. w  e.  C  Pred ( R ,  A ,  X
)  C_  dom  w )
26 ssiun 4217 . . . . . . . 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 5054 . . . . . . 7  |-  dom  U. C  =  U_ w  e.  C  dom  w
2927, 28syl6sseqr 3408 . . . . . 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 2846 . 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 965    = wceq 1369   E.wex 1586    e. wcel 1756   {cab 2429   A.wral 2720   E.wrex 2721    C_ wss 3333   U.cuni 4096   U_ciun 4176    Fr wfr 4681   Se wse 4682   dom cdm 4845    |` cres 4847    Fn wfn 5418   ` cfv 5423  (class class class)co 6096   Predcpred 27629
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 2423
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431  df-ov 6099  df-pred 27630
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator