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Theorem bnj1522 28370
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
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
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 2285 . 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 28369 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 936    = wceq 1624    e. wcel 1685   {cab 2271    =/= wne 2448   A.wral 2545   E.wrex 2546   {crab 2549    C_ wss 3154   <.cop 3645   U.cuni 3829   class class class wbr 4025    |` cres 4691    Fn wfn 5217   ` cfv 5222    predc-bnj14 27981    FrSe w-bnj15 27985
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7302  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-iota 6253  df-1o 6475  df-bnj17 27980  df-bnj14 27982  df-bnj13 27984  df-bnj15 27986  df-bnj18 27988  df-bnj19 27990
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