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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 990 . . . . . . 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 3050 . . . . . . . . . . . 12  |-  x  e. 
_V
54elon 5435 . . . . . . . . . . 11  |-  ( x  e.  On  <->  Ord  x )
6 ordsson 6621 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  x  C_  On )
7 ordtr 5440 . . . . . . . . . . . . 13  |-  ( Ord  x  ->  Tr  x
)
86, 7jca 535 . . . . . . . . . . . 12  |-  ( Ord  x  ->  ( x  C_  On  /\  Tr  x
) )
9 epweon 6615 . . . . . . . . . . . . . . . 16  |-  _E  We  On
10 wess 4824 . . . . . . . . . . . . . . . 16  |-  ( x 
C_  On  ->  (  _E  We  On  ->  _E  We  x ) )
119, 10mpi 20 . . . . . . . . . . . . . . 15  |-  ( x 
C_  On  ->  _E  We  x )
1211anim2i 573 . . . . . . . . . . . . . 14  |-  ( ( Tr  x  /\  x  C_  On )  ->  ( Tr  x  /\  _E  We  x ) )
1312ancoms 455 . . . . . . . . . . . . 13  |-  ( ( x  C_  On  /\  Tr  x )  ->  ( Tr  x  /\  _E  We  x ) )
14 df-ord 5429 . . . . . . . . . . . . 13  |-  ( Ord  x  <->  ( Tr  x  /\  _E  We  x ) )
1513, 14sylibr 216 . . . . . . . . . . . 12  |-  ( ( x  C_  On  /\  Tr  x )  ->  Ord  x )
168, 15impbii 191 . . . . . . . . . . 11  |-  ( Ord  x  <->  ( x  C_  On  /\  Tr  x ) )
17 ssel2 3429 . . . . . . . . . . . . . . 15  |-  ( ( x  C_  On  /\  y  e.  x )  ->  y  e.  On )
18 predon 6623 . . . . . . . . . . . . . . . 16  |-  ( y  e.  On  ->  Pred (  _E  ,  On ,  y )  =  y )
1918sseq1d 3461 . . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . 13  |-  ( x 
C_  On  ->  ( A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x 
<-> 
A. y  e.  x  y  C_  x ) )
22 dftr3 4504 . . . . . . . . . . . . 13  |-  ( Tr  x  <->  A. y  e.  x  y  C_  x )
2321, 22syl6rbbr 268 . . . . . . . . . . . 12  |-  ( x 
C_  On  ->  ( Tr  x  <->  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x ) )
2423pm5.32i 643 . . . . . . . . . . 11  |-  ( ( x  C_  On  /\  Tr  x )  <->  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )
)
255, 16, 243bitri 275 . . . . . . . . . 10  |-  ( x  e.  On  <->  ( x  C_  On  /\  A. y  e.  x  Pred (  _E  ,  On ,  y )  C_  x )
)
2625anbi1i 702 . . . . . . . . 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 5451 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  y  e.  x )  ->  y  e.  On )
2818reseq2d 5108 . . . . . . . . . . . . . 14  |-  ( y  e.  On  ->  (
f  |`  Pred (  _E  ,  On ,  y )
)  =  ( f  |`  y ) )
2928fveq2d 5874 . . . . . . . . . . . . 13  |-  ( y  e.  On  ->  ( F `  ( f  |` 
Pred (  _E  ,  On ,  y )
) )  =  ( F `  ( f  |`  y ) ) )
3029eqeq2d 2463 . . . . . . . . . . . 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 2826 . . . . . . . . . 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 643 . . . . . . . . 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 255 . . . . . . . 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 701 . . . . . . 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 807 . . . . . . 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 275 . . . . . 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 1720 . . . . 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 2745 . . . . 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 256 . . . 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 2569 . . 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 4210 . 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 2479 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 188    /\ wa 371    /\ w3a 986    = wceq 1446   E.wex 1665    e. wcel 1889   {cab 2439   A.wral 2739   E.wrex 2740    C_ wss 3406   U.cuni 4201   Tr wtr 4500    _E cep 4746    We wwe 4795    |` cres 4839   Predcpred 5382   Ord word 5425   Oncon0 5426    Fn wfn 5580   ` cfv 5585  wrecscwrecs 7032  recscrecs 7094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-tr 4501  df-eprel 4748  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-iota 5549  df-fv 5593  df-wrecs 7033  df-recs 7095
This theorem is referenced by:  recsfval  7104  tfrlem9  7108  dfrdg2  30454  dfrecs2  30729
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