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Theorem rdgprc 25414
Description: The value of the recursive definition generator when 
I is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
rdgprc  |-  ( -.  I  e.  _V  ->  rec ( F ,  I
)  =  rec ( F ,  (/) ) )

Proof of Theorem rdgprc
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5720 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  (/) ) )
2 fveq2 5720 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  (/) ) )
31, 2eqeq12d 2449 . . . . . 6  |-  ( z  =  (/)  ->  ( ( rec ( F ,  I ) `  z
)  =  ( rec ( F ,  (/) ) `  z )  <->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) ) )
43imbi2d 308 . . . . 5  |-  ( z  =  (/)  ->  ( ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z ) )  <->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `
 (/) )  =  ( rec ( F ,  (/) ) `  (/) ) ) ) )
5 fveq2 5720 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  y ) )
6 fveq2 5720 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  y ) )
75, 6eqeq12d 2449 . . . . . 6  |-  ( z  =  y  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) ) )
87imbi2d 308 . . . . 5  |-  ( z  =  y  ->  (
( -.  I  e. 
_V  ->  ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z ) )  <->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `
 y )  =  ( rec ( F ,  (/) ) `  y
) ) ) )
9 fveq2 5720 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  I ) `  suc  y ) )
10 fveq2 5720 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  (/) ) `  z
)  =  ( rec ( F ,  (/) ) `  suc  y ) )
119, 10eqeq12d 2449 . . . . . 6  |-  ( z  =  suc  y  -> 
( ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
) ) )
1211imbi2d 308 . . . . 5  |-  ( z  =  suc  y  -> 
( ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `
 z )  =  ( rec ( F ,  (/) ) `  z
) )  <->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
) ) ) )
13 fveq2 5720 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  x ) )
14 fveq2 5720 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  x ) )
1513, 14eqeq12d 2449 . . . . . 6  |-  ( z  =  x  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) ) )
1615imbi2d 308 . . . . 5  |-  ( z  =  x  ->  (
( -.  I  e. 
_V  ->  ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z ) )  <->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `
 x )  =  ( rec ( F ,  (/) ) `  x
) ) ) )
17 rdgprc0 25413 . . . . . 6  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
18 0ex 4331 . . . . . . 7  |-  (/)  e.  _V
1918rdg0 6671 . . . . . 6  |-  ( rec ( F ,  (/) ) `  (/) )  =  (/)
2017, 19syl6eqr 2485 . . . . 5  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) )
21 fveq2 5720 . . . . . . 7  |-  ( ( rec ( F ,  I ) `  y
)  =  ( rec ( F ,  (/) ) `  y )  ->  ( F `  ( rec ( F ,  I
) `  y )
)  =  ( F `
 ( rec ( F ,  (/) ) `  y ) ) )
22 rdgsuc 6674 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  I
) `  suc  y )  =  ( F `  ( rec ( F ,  I ) `  y
) ) )
23 rdgsuc 6674 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  (/) ) `  suc  y )  =  ( F `  ( rec ( F ,  (/) ) `  y ) ) )
2422, 23eqeq12d 2449 . . . . . . 7  |-  ( y  e.  On  ->  (
( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
)  <->  ( F `  ( rec ( F ,  I ) `  y
) )  =  ( F `  ( rec ( F ,  (/) ) `  y )
) ) )
2521, 24syl5ibr 213 . . . . . 6  |-  ( y  e.  On  ->  (
( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y )  ->  ( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y ) ) )
2625imim2d 50 . . . . 5  |-  ( y  e.  On  ->  (
( -.  I  e. 
_V  ->  ( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) )  ->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
) ) ) )
27 r19.21v 2785 . . . . . 6  |-  ( A. y  e.  z  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) )  <->  ( -.  I  e.  _V  ->  A. y  e.  z  ( rec ( F ,  I ) `
 y )  =  ( rec ( F ,  (/) ) `  y
) ) )
28 limord 4632 . . . . . . . . 9  |-  ( Lim  z  ->  Ord  z )
29 ordsson 4762 . . . . . . . . 9  |-  ( Ord  z  ->  z  C_  On )
30 rdgfnon 6668 . . . . . . . . . 10  |-  rec ( F ,  I )  Fn  On
31 rdgfnon 6668 . . . . . . . . . 10  |-  rec ( F ,  (/) )  Fn  On
32 fvreseq 5825 . . . . . . . . . 10  |-  ( ( ( rec ( F ,  I )  Fn  On  /\  rec ( F ,  (/) )  Fn  On )  /\  z  C_  On )  ->  (
( rec ( F ,  I )  |`  z )  =  ( rec ( F ,  (/) )  |`  z )  <->  A. y  e.  z  ( rec ( F ,  I ) `  y
)  =  ( rec ( F ,  (/) ) `  y )
) )
3330, 31, 32mpanl12 664 . . . . . . . . 9  |-  ( z 
C_  On  ->  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  <->  A. y  e.  z  ( rec ( F ,  I ) `
 y )  =  ( rec ( F ,  (/) ) `  y
) ) )
3428, 29, 333syl 19 . . . . . . . 8  |-  ( Lim  z  ->  ( ( rec ( F ,  I
)  |`  z )  =  ( rec ( F ,  (/) )  |`  z
)  <->  A. y  e.  z  ( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) ) )
35 rneq 5087 . . . . . . . . . . 11  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  ran  ( rec ( F ,  I )  |`  z )  =  ran  ( rec ( F ,  (/) )  |`  z )
)
36 df-ima 4883 . . . . . . . . . . 11  |-  ( rec ( F ,  I
) " z )  =  ran  ( rec ( F ,  I
)  |`  z )
37 df-ima 4883 . . . . . . . . . . 11  |-  ( rec ( F ,  (/) ) " z )  =  ran  ( rec ( F ,  (/) )  |`  z )
3835, 36, 373eqtr4g 2492 . . . . . . . . . 10  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  -> 
( rec ( F ,  I ) "
z )  =  ( rec ( F ,  (/) ) " z ) )
3938unieqd 4018 . . . . . . . . 9  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) )
40 vex 2951 . . . . . . . . . 10  |-  z  e. 
_V
41 rdglim 6676 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  I
) `  z )  =  U. ( rec ( F ,  I ) " z ) )
42 rdglim 6676 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  (/) ) `  z )  =  U. ( rec ( F ,  (/) ) "
z ) )
4341, 42eqeq12d 2449 . . . . . . . . . 10  |-  ( ( z  e.  _V  /\  Lim  z )  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) ) )
4440, 43mpan 652 . . . . . . . . 9  |-  ( Lim  z  ->  ( ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  (/) ) `  z )  <->  U. ( rec ( F ,  I
) " z )  =  U. ( rec ( F ,  (/) ) " z ) ) )
4539, 44syl5ibr 213 . . . . . . . 8  |-  ( Lim  z  ->  ( ( rec ( F ,  I
)  |`  z )  =  ( rec ( F ,  (/) )  |`  z
)  ->  ( rec ( F ,  I ) `
 z )  =  ( rec ( F ,  (/) ) `  z
) ) )
4634, 45sylbird 227 . . . . . . 7  |-  ( Lim  z  ->  ( A. y  e.  z  ( rec ( F ,  I
) `  y )  =  ( rec ( F ,  (/) ) `  y )  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  (/) ) `  z ) ) )
4746imim2d 50 . . . . . 6  |-  ( Lim  z  ->  ( ( -.  I  e.  _V  ->  A. y  e.  z  ( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) )  ->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  z
)  =  ( rec ( F ,  (/) ) `  z )
) ) )
4827, 47syl5bi 209 . . . . 5  |-  ( Lim  z  ->  ( A. y  e.  z  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) )  ->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  z
)  =  ( rec ( F ,  (/) ) `  z )
) ) )
494, 8, 12, 16, 20, 26, 48tfinds 4831 . . . 4  |-  ( x  e.  On  ->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) ) )
5049com12 29 . . 3  |-  ( -.  I  e.  _V  ->  ( x  e.  On  ->  ( rec ( F ,  I ) `  x
)  =  ( rec ( F ,  (/) ) `  x )
) )
5150ralrimiv 2780 . 2  |-  ( -.  I  e.  _V  ->  A. x  e.  On  ( rec ( F ,  I
) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
52 eqfnfv 5819 . . 3  |-  ( ( rec ( F ,  I )  Fn  On  /\ 
rec ( F ,  (/) )  Fn  On )  ->  ( rec ( F ,  I )  =  rec ( F ,  (/) )  <->  A. x  e.  On  ( rec ( F ,  I ) `  x
)  =  ( rec ( F ,  (/) ) `  x )
) )
5330, 31, 52mp2an 654 . 2  |-  ( rec ( F ,  I
)  =  rec ( F ,  (/) )  <->  A. x  e.  On  ( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
5451, 53sylibr 204 1  |-  ( -.  I  e.  _V  ->  rec ( F ,  I
)  =  rec ( F ,  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    C_ wss 3312   (/)c0 3620   U.cuni 4007   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575   ran crn 4871    |` cres 4872   "cima 4873    Fn wfn 5441   ` cfv 5446   reccrdg 6659
This theorem is referenced by:  dfrdg3  25416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-recs 6625  df-rdg 6660
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