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Theorem rdgprc 25169
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 5662 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  (/) ) )
2 fveq2 5662 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  (/) ) )
31, 2eqeq12d 2395 . . . . . 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 5662 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  y ) )
6 fveq2 5662 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  y ) )
75, 6eqeq12d 2395 . . . . . 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 5662 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  I ) `  suc  y ) )
10 fveq2 5662 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  (/) ) `  z
)  =  ( rec ( F ,  (/) ) `  suc  y ) )
119, 10eqeq12d 2395 . . . . . 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 5662 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  x ) )
14 fveq2 5662 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  x ) )
1513, 14eqeq12d 2395 . . . . . 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 25168 . . . . . 6  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
18 0ex 4274 . . . . . . 7  |-  (/)  e.  _V
1918rdg0 6609 . . . . . 6  |-  ( rec ( F ,  (/) ) `  (/) )  =  (/)
2017, 19syl6eqr 2431 . . . . 5  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) )
21 fveq2 5662 . . . . . . 7  |-  ( ( rec ( F ,  I ) `  y
)  =  ( rec ( F ,  (/) ) `  y )  ->  ( F `  ( rec ( F ,  I
) `  y )
)  =  ( F `
 ( rec ( F ,  (/) ) `  y ) ) )
22 rdgsuc 6612 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  I
) `  suc  y )  =  ( F `  ( rec ( F ,  I ) `  y
) ) )
23 rdgsuc 6612 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  (/) ) `  suc  y )  =  ( F `  ( rec ( F ,  (/) ) `  y ) ) )
2422, 23eqeq12d 2395 . . . . . . 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 2730 . . . . . 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 4575 . . . . . . . . 9  |-  ( Lim  z  ->  Ord  z )
29 ordsson 4704 . . . . . . . . 9  |-  ( Ord  z  ->  z  C_  On )
30 rdgfnon 6606 . . . . . . . . . 10  |-  rec ( F ,  I )  Fn  On
31 rdgfnon 6606 . . . . . . . . . 10  |-  rec ( F ,  (/) )  Fn  On
32 fvreseq 5766 . . . . . . . . . 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 5029 . . . . . . . . . . 11  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  ran  ( rec ( F ,  I )  |`  z )  =  ran  ( rec ( F ,  (/) )  |`  z )
)
36 df-ima 4825 . . . . . . . . . . 11  |-  ( rec ( F ,  I
) " z )  =  ran  ( rec ( F ,  I
)  |`  z )
37 df-ima 4825 . . . . . . . . . . 11  |-  ( rec ( F ,  (/) ) " z )  =  ran  ( rec ( F ,  (/) )  |`  z )
3835, 36, 373eqtr4g 2438 . . . . . . . . . 10  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  -> 
( rec ( F ,  I ) "
z )  =  ( rec ( F ,  (/) ) " z ) )
3938unieqd 3962 . . . . . . . . 9  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) )
40 vex 2896 . . . . . . . . . 10  |-  z  e. 
_V
41 rdglim 6614 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  I
) `  z )  =  U. ( rec ( F ,  I ) " z ) )
42 rdglim 6614 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  (/) ) `  z )  =  U. ( rec ( F ,  (/) ) "
z ) )
4341, 42eqeq12d 2395 . . . . . . . . . 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 4773 . . . 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 2725 . 2  |-  ( -.  I  e.  _V  ->  A. x  e.  On  ( rec ( F ,  I
) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
52 eqfnfv 5760 . . 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 1649    e. wcel 1717   A.wral 2643   _Vcvv 2893    C_ wss 3257   (/)c0 3565   U.cuni 3951   Ord word 4515   Oncon0 4516   Lim wlim 4517   suc csuc 4518   ran crn 4813    |` cres 4814   "cima 4815    Fn wfn 5383   ` cfv 5388   reccrdg 6597
This theorem is referenced by:  dfrdg3  25171
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2362  ax-rep 4255  ax-sep 4265  ax-nul 4273  ax-pow 4312  ax-pr 4338  ax-un 4635
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 2236  df-mo 2237  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2506  df-ne 2546  df-ral 2648  df-rex 2649  df-reu 2650  df-rab 2652  df-v 2895  df-sbc 3099  df-csb 3189  df-dif 3260  df-un 3262  df-in 3264  df-ss 3271  df-pss 3273  df-nul 3566  df-if 3677  df-pw 3738  df-sn 3757  df-pr 3758  df-tp 3759  df-op 3760  df-uni 3952  df-iun 4031  df-br 4148  df-opab 4202  df-mpt 4203  df-tr 4238  df-eprel 4429  df-id 4433  df-po 4438  df-so 4439  df-fr 4476  df-we 4478  df-ord 4519  df-on 4520  df-lim 4521  df-suc 4522  df-om 4780  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-iota 5352  df-fun 5390  df-fn 5391  df-f 5392  df-f1 5393  df-fo 5394  df-f1o 5395  df-fv 5396  df-recs 6563  df-rdg 6598
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