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Theorem dfrecs3 7096
Description: The old definition of transfinite recursion. This version is preferred for developement, as it demonstrates the properties of transfinite recursion without relying on well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.)
Assertion
Ref Expression
dfrecs3  |- recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
Distinct variable group:    f, F, x, y

Proof of Theorem dfrecs3
StepHypRef Expression
1 df-recs 7095 . 2  |- recs ( F )  = wrecs (  _E  ,  On ,  F
)
2 df-wrecs 7033 . 2  |- wrecs (  _E  ,  On ,  F
)  =  U. {
f  |  E. x
( f  Fn  x  /\  ( x  C_  On  /\ 
A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) }
3 3anass 986 . . . . . . 7  |-  ( ( f  Fn  x  /\  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) )  <-> 
( f  Fn  x  /\  ( ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) ) )
4 vex 3084 . . . . . . . . . . . 12  |-  x  e. 
_V
54elon 5448 . . . . . . . . . . 11  |-  ( x  e.  On  <->  Ord  x )
6 ordsson 6627 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  x  C_  On )
7 ordtr 5453 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
86, 7jca 534 . . . . . . . . . . . 12  |-  ( Ord  x  ->  ( x  C_  On  /\  Tr  x
) )
9 epweon 6621 . . . . . . . . . . . . . . . 16  |-  _E  We  On
10 wess 4837 . . . . . . . . . . . . . . . 16  |-  ( x 
C_  On  ->  (  _E  We  On  ->  _E  We  x ) )
119, 10mpi 21 . . . . . . . . . . . . . . 15  |-  ( x 
C_  On  ->  _E  We  x )
1211anim2i 571 . . . . . . . . . . . . . 14  |-  ( ( Tr  x  /\  x  C_  On )  ->  ( Tr  x  /\  _E  We  x ) )
1312ancoms 454 . . . . . . . . . . . . 13  |-  ( ( x  C_  On  /\  Tr  x )  ->  ( Tr  x  /\  _E  We  x ) )
14 df-ord 5442 . . . . . . . . . . . . 13  |-  ( Ord  x  <->  ( Tr  x  /\  _E  We  x ) )
1513, 14sylibr 215 . . . . . . . . . . . 12  |-  ( ( x  C_  On  /\  Tr  x )  ->  Ord  x )
168, 15impbii 190 . . . . . . . . . . 11  |-  ( Ord  x  <->  ( x  C_  On  /\  Tr  x ) )
17 ssel2 3459 . . . . . . . . . . . . . . 15  |-  ( ( x  C_  On  /\  y  e.  x )  ->  y  e.  On )
18 predon 6629 . . . . . . . . . . . . . . . 16  |-  ( y  e.  On  ->  Pred (  _E  ,  On ,  y )  =  y )
1918sseq1d 3491 . . . . . . . . . . . . . . 15  |-  ( y  e.  On  ->  ( Pred (  _E  ,  On ,  y )  C_  x 
<->  y  C_  x )
)
2017, 19syl 17 . . . . . . . . . . . . . 14  |-  ( ( x  C_  On  /\  y  e.  x )  ->  ( Pred (  _E  ,  On ,  y )  C_  x 
<->  y  C_  x )
)
2120ralbidva 2861 . . . . . . . . . . . . 13  |-  ( x 
C_  On  ->  ( A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x 
<-> 
A. y  e.  x  y  C_  x ) )
22 dftr3 4519 . . . . . . . . . . . . 13  |-  ( Tr  x  <->  A. y  e.  x  y  C_  x )
2321, 22syl6rbbr 267 . . . . . . . . . . . 12  |-  ( x 
C_  On  ->  ( Tr  x  <->  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x ) )
2423pm5.32i 641 . . . . . . . . . . 11  |-  ( ( x  C_  On  /\  Tr  x )  <->  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )
)
255, 16, 243bitri 274 . . . . . . . . . 10  |-  ( x  e.  On  <->  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )
)
2625anbi1i 699 . . . . . . . . 9  |-  ( ( x  e.  On  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) )  <-> 
( ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) )
27 onelon 5464 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
2818reseq2d 5121 . . . . . . . . . . . . . 14  |-  ( y  e.  On  ->  (
f  |`  Pred (  _E  ,  On ,  y )
)  =  ( f  |`  y ) )
2928fveq2d 5882 . . . . . . . . . . . . 13  |-  ( y  e.  On  ->  ( F `  ( f  |` 
Pred (  _E  ,  On ,  y )
) )  =  ( F `  ( f  |`  y ) ) )
3029eqeq2d 2436 . . . . . . . . . . . 12  |-  ( y  e.  On  ->  (
( f `  y
)  =  ( F `
 ( f  |`  Pred (  _E  ,  On ,  y ) ) )  <->  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
3127, 30syl 17 . . . . . . . . . . 11  |-  ( ( x  e.  On  /\  y  e.  x )  ->  ( ( f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y )
) )  <->  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
3231ralbidva 2861 . . . . . . . . . 10  |-  ( x  e.  On  ->  ( A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  Pred (  _E  ,  On ,  y ) ) )  <->  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )
3332pm5.32i 641 . . . . . . . . 9  |-  ( ( x  e.  On  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) )  <-> 
( x  e.  On  /\ 
A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )
3426, 33bitr3i 254 . . . . . . . 8  |-  ( ( ( x  C_  On  /\ 
A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) )  <-> 
( x  e.  On  /\ 
A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )
3534anbi2i 698 . . . . . . 7  |-  ( ( f  Fn  x  /\  ( ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) )  <->  ( f  Fn  x  /\  ( x  e.  On  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) ) ) )
36 an12 804 . . . . . . 7  |-  ( ( f  Fn  x  /\  ( x  e.  On  /\ 
A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) )  <->  ( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  y
) ) ) ) )
373, 35, 363bitri 274 . . . . . 6  |-  ( ( f  Fn  x  /\  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) )  <-> 
( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) ) )
3837exbii 1712 . . . . 5  |-  ( E. x ( f  Fn  x  /\  ( x 
C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  Pred (  _E  ,  On ,  y ) ) ) )  <->  E. x
( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) ) )
39 df-rex 2781 . . . . 5  |-  ( E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) )  <->  E. x
( x  e.  On  /\  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) ) ) )
4038, 39bitr4i 255 . . . 4  |-  ( E. x ( f  Fn  x  /\  ( x 
C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  Pred (  _E  ,  On ,  y ) ) ) )  <->  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) )
4140abbii 2556 . . 3  |-  { f  |  E. x ( f  Fn  x  /\  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) }  =  { f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
4241unieqi 4225 . 2  |-  U. {
f  |  E. x
( f  Fn  x  /\  ( x  C_  On  /\ 
A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )  /\  A. y  e.  x  (
f `  y )  =  ( F `  ( f  |`  Pred (  _E  ,  On ,  y ) ) ) ) }  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
431, 2, 423eqtri 2455 1  |- recs ( F )  =  U. {
f  |  E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) ) }
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437   E.wex 1659    e. wcel 1868   {cab 2407   A.wral 2775   E.wrex 2776    C_ wss 3436   U.cuni 4216   Tr wtr 4515    _E cep 4759    We wwe 4808    |` cres 4852   Predcpred 5395   Ord word 5438   Oncon0 5439    Fn wfn 5593   ` cfv 5598  wrecscwrecs 7032  recscrecs 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-tr 4516  df-eprel 4761  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-iota 5562  df-fv 5606  df-wrecs 7033  df-recs 7095
This theorem is referenced by:  recsfval  7104  tfrlem9  7108  dfrdg2  30437  dfrecs2  30710
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