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Theorem rdgprc 24222
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 5541 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  (/) ) )
2 fveq2 5541 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  (/) ) )
31, 2eqeq12d 2310 . . . . . 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 5541 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  y ) )
6 fveq2 5541 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  y ) )
75, 6eqeq12d 2310 . . . . . 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 5541 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  I ) `  suc  y ) )
10 fveq2 5541 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  (/) ) `  z
)  =  ( rec ( F ,  (/) ) `  suc  y ) )
119, 10eqeq12d 2310 . . . . . 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 5541 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  x ) )
14 fveq2 5541 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  x ) )
1513, 14eqeq12d 2310 . . . . . 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 24221 . . . . . 6  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
18 0ex 4166 . . . . . . 7  |-  (/)  e.  _V
1918rdg0 6450 . . . . . 6  |-  ( rec ( F ,  (/) ) `  (/) )  =  (/)
2017, 19syl6eqr 2346 . . . . 5  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) )
21 fveq2 5541 . . . . . . 7  |-  ( ( rec ( F ,  I ) `  y
)  =  ( rec ( F ,  (/) ) `  y )  ->  ( F `  ( rec ( F ,  I
) `  y )
)  =  ( F `
 ( rec ( F ,  (/) ) `  y ) ) )
22 rdgsuc 6453 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  I
) `  suc  y )  =  ( F `  ( rec ( F ,  I ) `  y
) ) )
23 rdgsuc 6453 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  (/) ) `  suc  y )  =  ( F `  ( rec ( F ,  (/) ) `  y ) ) )
2422, 23eqeq12d 2310 . . . . . . 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 2643 . . . . . 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 4467 . . . . . . . . 9  |-  ( Lim  z  ->  Ord  z )
29 ordsson 4597 . . . . . . . . 9  |-  ( Ord  z  ->  z  C_  On )
30 rdgfnon 6447 . . . . . . . . . 10  |-  rec ( F ,  I )  Fn  On
31 rdgfnon 6447 . . . . . . . . . 10  |-  rec ( F ,  (/) )  Fn  On
32 fvreseq 5644 . . . . . . . . . 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 4920 . . . . . . . . . . 11  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  ran  ( rec ( F ,  I )  |`  z )  =  ran  ( rec ( F ,  (/) )  |`  z )
)
36 df-ima 4718 . . . . . . . . . . 11  |-  ( rec ( F ,  I
) " z )  =  ran  ( rec ( F ,  I
)  |`  z )
37 df-ima 4718 . . . . . . . . . . 11  |-  ( rec ( F ,  (/) ) " z )  =  ran  ( rec ( F ,  (/) )  |`  z )
3835, 36, 373eqtr4g 2353 . . . . . . . . . 10  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  -> 
( rec ( F ,  I ) "
z )  =  ( rec ( F ,  (/) ) " z ) )
3938unieqd 3854 . . . . . . . . 9  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) )
40 vex 2804 . . . . . . . . . 10  |-  z  e. 
_V
41 rdglim 6455 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  I
) `  z )  =  U. ( rec ( F ,  I ) " z ) )
42 rdglim 6455 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  (/) ) `  z )  =  U. ( rec ( F ,  (/) ) "
z ) )
4341, 42eqeq12d 2310 . . . . . . . . . 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 4666 . . . 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 2638 . 2  |-  ( -.  I  e.  _V  ->  A. x  e.  On  ( rec ( F ,  I
) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
52 eqfnfv 5638 . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   (/)c0 3468   U.cuni 3843   Ord word 4407   Oncon0 4408   Lim wlim 4409   suc csuc 4410   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   ` cfv 5271   reccrdg 6438
This theorem is referenced by:  dfrdg3  24224
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-recs 6404  df-rdg 6439
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