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Theorem bnj60 34261
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 34259 . . . 4  |-  A. g  e.  C  Fun  g
5 eqid 2457 . . . . . . . 8  |-  ( dom  g  i^i  dom  h
)  =  ( dom  g  i^i  dom  h
)
61, 2, 3, 5bnj1311 34223 . . . . . . 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 1198 . . . . . 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 2869 . . . . 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 2871 . . . 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 236 . . . . 5  |-  ( A. g  e.  C  Fun  g 
<-> 
A. g  e.  C  Fun  g )
11 biid 236 . . . . 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 34033 . . . 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 663 . . 3  |-  ( R 
FrSe  A  ->  Fun  U. C )
14 bnj60.4 . . . 4  |-  F  = 
U. C
1514funeqi 5614 . . 3  |-  ( Fun 
F  <->  Fun  U. C )
1613, 15sylibr 212 . 2  |-  ( R 
FrSe  A  ->  Fun  F
)
171, 2, 3, 14bnj1498 34260 . 2  |-  ( R 
FrSe  A  ->  dom  F  =  A )
1816, 17bnj1422 34039 1  |-  ( R 
FrSe  A  ->  F  Fn  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   {cab 2442   A.wral 2807   E.wrex 2808    i^i cin 3470    C_ wss 3471   <.cop 4038   U.cuni 4251   dom cdm 5008    |` cres 5010   Fun wfun 5588    Fn wfn 5589   ` cfv 5594    predc-bnj14 33883    FrSe w-bnj15 33887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-reg 8036  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-om 6700  df-1o 7148  df-bnj17 33882  df-bnj14 33884  df-bnj13 33886  df-bnj15 33888  df-bnj18 33890  df-bnj19 33892
This theorem is referenced by:  bnj1501  34266  bnj1523  34270
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