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Theorem rdgprc 24151
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 5525 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  (/) ) )
2 fveq2 5525 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  (/) ) )
31, 2eqeq12d 2297 . . . . . 6  |-  ( z  =  (/)  ->  ( ( rec ( F ,  I ) `  z
)  =  ( rec ( F ,  (/) ) `  z )  <->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) ) )
43imbi2d 307 . . . . 5  |-  ( z  =  (/)  ->  ( ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z ) )  <->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `
 (/) )  =  ( rec ( F ,  (/) ) `  (/) ) ) ) )
5 fveq2 5525 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  y ) )
6 fveq2 5525 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  y ) )
75, 6eqeq12d 2297 . . . . . 6  |-  ( z  =  y  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) ) )
87imbi2d 307 . . . . 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 5525 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  I ) `  suc  y ) )
10 fveq2 5525 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  (/) ) `  z
)  =  ( rec ( F ,  (/) ) `  suc  y ) )
119, 10eqeq12d 2297 . . . . . 6  |-  ( z  =  suc  y  -> 
( ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
) ) )
1211imbi2d 307 . . . . 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 5525 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  x ) )
14 fveq2 5525 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  x ) )
1513, 14eqeq12d 2297 . . . . . 6  |-  ( z  =  x  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) ) )
1615imbi2d 307 . . . . 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 24150 . . . . . 6  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
18 0ex 4150 . . . . . . 7  |-  (/)  e.  _V
1918rdg0 6434 . . . . . 6  |-  ( rec ( F ,  (/) ) `  (/) )  =  (/)
2017, 19syl6eqr 2333 . . . . 5  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) )
21 fveq2 5525 . . . . . . 7  |-  ( ( rec ( F ,  I ) `  y
)  =  ( rec ( F ,  (/) ) `  y )  ->  ( F `  ( rec ( F ,  I
) `  y )
)  =  ( F `
 ( rec ( F ,  (/) ) `  y ) ) )
22 rdgsuc 6437 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  I
) `  suc  y )  =  ( F `  ( rec ( F ,  I ) `  y
) ) )
23 rdgsuc 6437 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  (/) ) `  suc  y )  =  ( F `  ( rec ( F ,  (/) ) `  y ) ) )
2422, 23eqeq12d 2297 . . . . . . 7  |-  ( y  e.  On  ->  (
( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
)  <->  ( F `  ( rec ( F ,  I ) `  y
) )  =  ( F `  ( rec ( F ,  (/) ) `  y )
) ) )
2521, 24syl5ibr 212 . . . . . 6  |-  ( y  e.  On  ->  (
( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y )  ->  ( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y ) ) )
2625imim2d 48 . . . . 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 2630 . . . . . 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 4451 . . . . . . . . 9  |-  ( Lim  z  ->  Ord  z )
29 ordsson 4581 . . . . . . . . 9  |-  ( Ord  z  ->  z  C_  On )
30 rdgfnon 6431 . . . . . . . . . 10  |-  rec ( F ,  I )  Fn  On
31 rdgfnon 6431 . . . . . . . . . 10  |-  rec ( F ,  (/) )  Fn  On
32 fvreseq 5628 . . . . . . . . . 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 663 . . . . . . . . 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 18 . . . . . . . 8  |-  ( Lim  z  ->  ( ( rec ( F ,  I
)  |`  z )  =  ( rec ( F ,  (/) )  |`  z
)  <->  A. y  e.  z  ( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) ) )
35 rneq 4904 . . . . . . . . . . 11  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  ran  ( rec ( F ,  I )  |`  z )  =  ran  ( rec ( F ,  (/) )  |`  z )
)
36 df-ima 4702 . . . . . . . . . . 11  |-  ( rec ( F ,  I
) " z )  =  ran  ( rec ( F ,  I
)  |`  z )
37 df-ima 4702 . . . . . . . . . . 11  |-  ( rec ( F ,  (/) ) " z )  =  ran  ( rec ( F ,  (/) )  |`  z )
3835, 36, 373eqtr4g 2340 . . . . . . . . . 10  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  -> 
( rec ( F ,  I ) "
z )  =  ( rec ( F ,  (/) ) " z ) )
3938unieqd 3838 . . . . . . . . 9  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) )
40 vex 2791 . . . . . . . . . 10  |-  z  e. 
_V
41 rdglim 6439 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  I
) `  z )  =  U. ( rec ( F ,  I ) " z ) )
42 rdglim 6439 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  (/) ) `  z )  =  U. ( rec ( F ,  (/) ) "
z ) )
4341, 42eqeq12d 2297 . . . . . . . . . 10  |-  ( ( z  e.  _V  /\  Lim  z )  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) ) )
4440, 43mpan 651 . . . . . . . . 9  |-  ( Lim  z  ->  ( ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  (/) ) `  z )  <->  U. ( rec ( F ,  I
) " z )  =  U. ( rec ( F ,  (/) ) " z ) ) )
4539, 44syl5ibr 212 . . . . . . . 8  |-  ( Lim  z  ->  ( ( rec ( F ,  I
)  |`  z )  =  ( rec ( F ,  (/) )  |`  z
)  ->  ( rec ( F ,  I ) `
 z )  =  ( rec ( F ,  (/) ) `  z
) ) )
4634, 45sylbird 226 . . . . . . 7  |-  ( Lim  z  ->  ( A. y  e.  z  ( rec ( F ,  I
) `  y )  =  ( rec ( F ,  (/) ) `  y )  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  (/) ) `  z ) ) )
4746imim2d 48 . . . . . 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 208 . . . . 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 4650 . . . 4  |-  ( x  e.  On  ->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) ) )
5049com12 27 . . 3  |-  ( -.  I  e.  _V  ->  ( x  e.  On  ->  ( rec ( F ,  I ) `  x
)  =  ( rec ( F ,  (/) ) `  x )
) )
5150ralrimiv 2625 . 2  |-  ( -.  I  e.  _V  ->  A. x  e.  On  ( rec ( F ,  I
) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
52 eqfnfv 5622 . . 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 653 . 2  |-  ( rec ( F ,  I
)  =  rec ( F ,  (/) )  <->  A. x  e.  On  ( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
5451, 53sylibr 203 1  |-  ( -.  I  e.  _V  ->  rec ( F ,  I
)  =  rec ( F ,  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   (/)c0 3455   U.cuni 3827   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   ` cfv 5255   reccrdg 6422
This theorem is referenced by:  dfrdg3  24153
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6388  df-rdg 6423
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