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Theorem bnj60 29883
Description: Well-founded recursion, part 1 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
bnj60.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj60.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj60.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj60.4  |-  F  = 
U. C
Assertion
Ref Expression
bnj60  |-  ( R 
FrSe  A  ->  F  Fn  A )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    R, d, f, x
Allowed substitution hints:    B( x, d)    C( x, f, d)    F( x, f, d)    Y( x, f, d)

Proof of Theorem bnj60
Dummy variables  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj60.1 . . . . 5  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj60.2 . . . . 5  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj60.3 . . . . 5  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
41, 2, 3bnj1497 29881 . . . 4  |-  A. g  e.  C  Fun  g
5 eqid 2453 . . . . . . . 8  |-  ( dom  g  i^i  dom  h
)  =  ( dom  g  i^i  dom  h
)
61, 2, 3, 5bnj1311 29845 . . . . . . 7  |-  ( ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C )  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) )
763expia 1211 . . . . . 6  |-  ( ( R  FrSe  A  /\  g  e.  C )  ->  ( h  e.  C  ->  ( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) ) )
87ralrimiv 2802 . . . . 5  |-  ( ( R  FrSe  A  /\  g  e.  C )  ->  A. h  e.  C  ( g  |`  ( dom  g  i^i  dom  h
) )  =  ( h  |`  ( dom  g  i^i  dom  h )
) )
98ralrimiva 2804 . . . 4  |-  ( R 
FrSe  A  ->  A. g  e.  C  A. h  e.  C  ( g  |`  ( dom  g  i^i 
dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )
10 biid 240 . . . . 5  |-  ( A. g  e.  C  Fun  g 
<-> 
A. g  e.  C  Fun  g )
11 biid 240 . . . . 5  |-  ( ( A. g  e.  C  Fun  g  /\  A. g  e.  C  A. h  e.  C  ( g  |`  ( dom  g  i^i 
dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )  <-> 
( A. g  e.  C  Fun  g  /\  A. g  e.  C  A. h  e.  C  (
g  |`  ( dom  g  i^i  dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) ) )
1210, 5, 11bnj1383 29655 . . . 4  |-  ( ( A. g  e.  C  Fun  g  /\  A. g  e.  C  A. h  e.  C  ( g  |`  ( dom  g  i^i 
dom  h ) )  =  ( h  |`  ( dom  g  i^i  dom  h ) ) )  ->  Fun  U. C )
134, 9, 12sylancr 670 . . 3  |-  ( R 
FrSe  A  ->  Fun  U. C )
14 bnj60.4 . . . 4  |-  F  = 
U. C
1514funeqi 5605 . . 3  |-  ( Fun 
F  <->  Fun  U. C )
1613, 15sylibr 216 . 2  |-  ( R 
FrSe  A  ->  Fun  F
)
171, 2, 3, 14bnj1498 29882 . 2  |-  ( R 
FrSe  A  ->  dom  F  =  A )
1816, 17bnj1422 29661 1  |-  ( R 
FrSe  A  ->  F  Fn  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    = wceq 1446    e. wcel 1889   {cab 2439   A.wral 2739   E.wrex 2740    i^i cin 3405    C_ wss 3406   <.cop 3976   U.cuni 4201   dom cdm 4837    |` cres 4839   Fun wfun 5579    Fn wfn 5580   ` cfv 5585    predc-bnj14 29505    FrSe w-bnj15 29509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-reg 8112  ax-inf2 8151
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-om 6698  df-1o 7187  df-bnj17 29504  df-bnj14 29506  df-bnj13 29508  df-bnj15 29510  df-bnj18 29512  df-bnj19 29514
This theorem is referenced by:  bnj1501  29888  bnj1523  29892
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