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Theorem bnj1500 29327
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 236 . 2  |-  ( ( R  FrSe  A  /\  x  e.  A )  <->  ( R  FrSe  A  /\  x  e.  A )
)
6 biid 236 . 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 236 . 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 29326 1  |-  ( R 
FrSe  A  ->  A. x  e.  A  ( F `  x )  =  ( G `  <. x ,  ( F  |`  pred ( x ,  A ,  R ) ) >.
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 972    = wceq 1403    e. wcel 1840   {cab 2385   A.wral 2751   E.wrex 2752    C_ wss 3411   <.cop 3975   U.cuni 4188   dom cdm 4940    |` cres 4942    Fn wfn 5518   ` cfv 5523    predc-bnj14 28943    FrSe w-bnj15 28947
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-reg 7970  ax-inf2 8009
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-fal 1409  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-om 6637  df-1o 7085  df-bnj17 28942  df-bnj14 28944  df-bnj13 28946  df-bnj15 28948  df-bnj18 28950  df-bnj19 28952
This theorem is referenced by:  bnj1523  29330
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