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Theorem bnj60 28359
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
Dummy variables  g  h are mutually distinct and distinct from all other variables.
Allowed substitution hints:    B( x, d)    C( x, f, d)    F( x, f, d)    Y( x, f, d)

Proof of Theorem bnj60
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 28357 . . . 4  |-  A. g  e.  C  Fun  g
5 eqid 2284 . . . . . . . 8  |-  ( dom  g  i^i  dom  h
)  =  ( dom  g  i^i  dom  h
)
61, 2, 3, 5bnj1311 28321 . . . . . . 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 1155 . . . . . 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 2626 . . . . 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 2627 . . . 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 28131 . . . 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 5241 . . 3  |-  ( Fun 
F  <->  Fun  U. C )
1613, 15sylibr 205 . 2  |-  ( R 
FrSe  A  ->  Fun  F
)
171, 2, 3, 14bnj1498 28358 . 2  |-  ( R 
FrSe  A  ->  dom  F  =  A )
1816, 17bnj1422 28137 1  |-  ( R 
FrSe  A  ->  F  Fn  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1624    e. wcel 1685   {cab 2270   A.wral 2544   E.wrex 2545    i^i cin 3152    C_ wss 3153   <.cop 3644   U.cuni 3828   dom cdm 4688    |` cres 4690   Fun wfun 5215    Fn wfn 5216   ` cfv 5221    predc-bnj14 27980    FrSe w-bnj15 27984
This theorem is referenced by:  bnj1501  28364  bnj1523  28368
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-reg 7301  ax-inf2 7337
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-iota 6252  df-1o 6474  df-bnj17 27979  df-bnj14 27981  df-bnj13 27983  df-bnj15 27985  df-bnj18 27987  df-bnj19 27989
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