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Theorem bnj1522 29147
Description: Well-founded recursion, part 3 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
bnj1522.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1522.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1522.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1522.4  |-  F  = 
U. C
Assertion
Ref Expression
bnj1522  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  ->  F  =  H )
Distinct variable groups:    A, d,
f, x    B, f    G, d, f, x    x, H    R, d, f, x    Y, d
Allowed substitution hints:    B( x, d)    C( x, f, d)    F( x, f, d)    H( f, d)    Y( x, f)

Proof of Theorem bnj1522
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bnj1522.1 . 2  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
2 bnj1522.2 . 2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
3 bnj1522.3 . 2  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
4 bnj1522.4 . 2  |-  F  = 
U. C
5 biid 228 . 2  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  <->  ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >.
) ) )
6 biid 228 . 2  |-  ( ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  <->  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >.
) )  /\  F  =/=  H ) )
7 biid 228 . 2  |-  ( ( ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x ) )  <->  ( (
( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x ) ) )
8 eqid 2404 . 2  |-  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  =  {
x  e.  A  | 
( F `  x
)  =/=  ( H `
 x ) }
9 biid 228 . 2  |-  ( ( ( ( ( R 
FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >.
) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x
) )  /\  y  e.  { x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }  /\  A. z  e. 
{ x  e.  A  |  ( F `  x )  =/=  ( H `  x ) }  -.  z R y )  <->  ( ( ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  /\  F  =/=  H )  /\  x  e.  A  /\  ( F `  x )  =/=  ( H `  x ) )  /\  y  e.  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  /\  A. z  e.  { x  e.  A  |  ( F `  x )  =/=  ( H `  x
) }  -.  z R y ) )
101, 2, 3, 4, 5, 6, 7, 8, 9bnj1523 29146 1  |-  ( ( R  FrSe  A  /\  H  Fn  A  /\  A. x  e.  A  ( H `  x )  =  ( G `  <. x ,  ( H  |`  pred ( x ,  A ,  R ) ) >. ) )  ->  F  =  H )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   <.cop 3777   U.cuni 3975   class class class wbr 4172    |` cres 4839    Fn wfn 5408   ` cfv 5413    predc-bnj14 28758    FrSe w-bnj15 28762
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-reg 7516  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-1o 6683  df-bnj17 28757  df-bnj14 28759  df-bnj13 28761  df-bnj15 28763  df-bnj18 28765  df-bnj19 28767
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