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Theorem bnj1522 28135
Description: Well-founded recursion, part 3 of 3. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1522.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1522.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1522.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1522.4  |-  F  = 
U. C
Assertion
Ref Expression
bnj1522  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  ->  F  =  H )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    x, H    R, d, f, x    Y, d
Allowed substitution hints:    B( x, d)    C( x, f, d)    F( x, f, d)    H( f, d)    Y( x, f)

Proof of Theorem bnj1522
StepHypRef Expression
1 bnj1522.1 . 2  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1522.2 . 2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1522.3 . 2  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1522.4 . 2  |-  F  = 
U. C
5 biid 229 . 2  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  <->  ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >.
) ) )
6 biid 229 . 2  |-  ( ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  <->  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >.
) )  /\  F  =/=  H ) )
7 biid 229 . 2  |-  ( ( ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x ) )  <->  ( (
( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x ) ) )
8 eqid 2256 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =/=  ( H `
 x ) }
9 biid 229 . 2  |-  ( ( ( ( ( R 
FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >.
) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x
) )  /\  y  e.  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }  /\  A. z  e. 
{ x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }  -.  z R y )  <->  ( ( ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x ) )  /\  y  e.  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  /\  A. z  e.  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  -.  z R y ) )
101, 2, 3, 4, 5, 6, 7, 8, 9bnj1523 28134 1  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  ->  F  =  H )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   {cab 2242    =/= wne 2419   A.wral 2516   E.wrex 2517   {crab 2520    C_ wss 3113   <.cop 3603   U.cuni 3787   class class class wbr 3983    |` cres 4649    Fn wfn 4654   ` cfv 4659    predc-bnj14 27746    FrSe w-bnj15 27750
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-9v 1632  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-reg 7260  ax-inf2 7296
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-iota 6211  df-1o 6433  df-bnj17 27745  df-bnj14 27747  df-bnj13 27749  df-bnj15 27751  df-bnj18 27753  df-bnj19 27755
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