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Theorem bnj60 29505
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 29503 . . . 4  |-  A. g  e.  C  Fun  g
5 eqid 2438 . . . . . . . 8  |-  ( dom  g  i^i  dom  h
)  =  ( dom  g  i^i  dom  h
)
61, 2, 3, 5bnj1311 29467 . . . . . . 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 1156 . . . . . 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 2790 . . . . 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 2791 . . . 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 229 . . . . 5  |-  ( A. g  e.  C  Fun  g 
<-> 
A. g  e.  C  Fun  g )
11 biid 229 . . . . 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 29277 . . . 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 646 . . 3  |-  ( R 
FrSe  A  ->  Fun  U. C )
14 bnj60.4 . . . 4  |-  F  = 
U. C
1514funeqi 5477 . . 3  |-  ( Fun 
F  <->  Fun  U. C )
1613, 15sylibr 205 . 2  |-  ( R 
FrSe  A  ->  Fun  F
)
171, 2, 3, 14bnj1498 29504 . 2  |-  ( R 
FrSe  A  ->  dom  F  =  A )
1816, 17bnj1422 29283 1  |-  ( R 
FrSe  A  ->  F  Fn  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708    i^i cin 3321    C_ wss 3322   <.cop 3819   U.cuni 4017   dom cdm 4881    |` cres 4883   Fun wfun 5451    Fn wfn 5452   ` cfv 5457    predc-bnj14 29126    FrSe w-bnj15 29130
This theorem is referenced by:  bnj1501  29510  bnj1523  29514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-reg 7563  ax-inf2 7599
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-1o 6727  df-bnj17 29125  df-bnj14 29127  df-bnj13 29129  df-bnj15 29131  df-bnj18 29133  df-bnj19 29135
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