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Theorem bnj1500 28471
Description: Well-founded recursion, part 2 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
bnj1500.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1500.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1500.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1500.4  |-  F  = 
U. C
Assertion
Ref Expression
bnj1500  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    R, d, f, x    Y, d
Allowed substitution hints:    B( x, d)    C( x, f, d)    F( x, f, d)    Y( x, f)

Proof of Theorem bnj1500
StepHypRef Expression
1 bnj1500.1 . 2  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1500.2 . 2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1500.3 . 2  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1500.4 . 2  |-  F  = 
U. C
5 biid 227 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  <->  ( R  FrSe  A  /\  x  e.  A )
)
6 biid 227 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  f  e.  C  /\  x  e.  dom  f )  <->  ( ( R  FrSe  A  /\  x  e.  A )  /\  f  e.  C  /\  x  e.  dom  f ) )
7 biid 227 . 2  |-  ( ( ( ( R  FrSe  A  /\  x  e.  A
)  /\  f  e.  C  /\  x  e.  dom  f )  /\  d  e.  B  /\  dom  f  =  d )  <->  ( (
( R  FrSe  A  /\  x  e.  A
)  /\  f  e.  C  /\  x  e.  dom  f )  /\  d  e.  B  /\  dom  f  =  d ) )
81, 2, 3, 4, 5, 6, 7bnj1501 28470 1  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544    C_ wss 3152   <.cop 3643   U.cuni 3827   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255    predc-bnj14 28086    FrSe w-bnj15 28090
This theorem is referenced by:  bnj1523  28474
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-reg 7306  ax-inf2 7342
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj17 28085  df-bnj14 28087  df-bnj13 28089  df-bnj15 28091  df-bnj18 28093  df-bnj19 28095
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