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Theorem rdgprc 30448
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 5881 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  (/) ) )
2 fveq2 5881 . . . . . . 7  |-  ( z  =  (/)  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  (/) ) )
31, 2eqeq12d 2444 . . . . . 6  |-  ( z  =  (/)  ->  ( ( rec ( F ,  I ) `  z
)  =  ( rec ( F ,  (/) ) `  z )  <->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) ) )
43imbi2d 317 . . . . 5  |-  ( z  =  (/)  ->  ( ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z ) )  <->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `
 (/) )  =  ( rec ( F ,  (/) ) `  (/) ) ) ) )
5 fveq2 5881 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  y ) )
6 fveq2 5881 . . . . . . 7  |-  ( z  =  y  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  y ) )
75, 6eqeq12d 2444 . . . . . 6  |-  ( z  =  y  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y ) ) )
87imbi2d 317 . . . . 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 5881 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  I ) `  suc  y ) )
10 fveq2 5881 . . . . . . 7  |-  ( z  =  suc  y  -> 
( rec ( F ,  (/) ) `  z
)  =  ( rec ( F ,  (/) ) `  suc  y ) )
119, 10eqeq12d 2444 . . . . . 6  |-  ( z  =  suc  y  -> 
( ( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
) ) )
1211imbi2d 317 . . . . 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 5881 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  I ) `  x ) )
14 fveq2 5881 . . . . . . 7  |-  ( z  =  x  ->  ( rec ( F ,  (/) ) `  z )  =  ( rec ( F ,  (/) ) `  x ) )
1513, 14eqeq12d 2444 . . . . . 6  |-  ( z  =  x  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <-> 
( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) ) )
1615imbi2d 317 . . . . 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 30447 . . . . . 6  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  (/) )
18 0ex 4556 . . . . . . 7  |-  (/)  e.  _V
1918rdg0 7150 . . . . . 6  |-  ( rec ( F ,  (/) ) `  (/) )  =  (/)
2017, 19syl6eqr 2481 . . . . 5  |-  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  (/) )  =  ( rec ( F ,  (/) ) `  (/) ) )
21 fveq2 5881 . . . . . . 7  |-  ( ( rec ( F ,  I ) `  y
)  =  ( rec ( F ,  (/) ) `  y )  ->  ( F `  ( rec ( F ,  I
) `  y )
)  =  ( F `
 ( rec ( F ,  (/) ) `  y ) ) )
22 rdgsuc 7153 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  I
) `  suc  y )  =  ( F `  ( rec ( F ,  I ) `  y
) ) )
23 rdgsuc 7153 . . . . . . . 8  |-  ( y  e.  On  ->  ( rec ( F ,  (/) ) `  suc  y )  =  ( F `  ( rec ( F ,  (/) ) `  y ) ) )
2422, 23eqeq12d 2444 . . . . . . 7  |-  ( y  e.  On  ->  (
( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y
)  <->  ( F `  ( rec ( F ,  I ) `  y
) )  =  ( F `  ( rec ( F ,  (/) ) `  y )
) ) )
2521, 24syl5ibr 224 . . . . . 6  |-  ( y  e.  On  ->  (
( rec ( F ,  I ) `  y )  =  ( rec ( F ,  (/) ) `  y )  ->  ( rec ( F ,  I ) `  suc  y )  =  ( rec ( F ,  (/) ) `  suc  y ) ) )
2625imim2d 54 . . . . 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 2827 . . . . . 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 5501 . . . . . . . . 9  |-  ( Lim  z  ->  Ord  z )
29 ordsson 6630 . . . . . . . . 9  |-  ( Ord  z  ->  z  C_  On )
30 rdgfnon 7147 . . . . . . . . . 10  |-  rec ( F ,  I )  Fn  On
31 rdgfnon 7147 . . . . . . . . . 10  |-  rec ( F ,  (/) )  Fn  On
32 fvreseq 5999 . . . . . . . . . 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 686 . . . . . . . . 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 5079 . . . . . . . . . . 11  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  ran  ( rec ( F ,  I )  |`  z )  =  ran  ( rec ( F ,  (/) )  |`  z )
)
36 df-ima 4866 . . . . . . . . . . 11  |-  ( rec ( F ,  I
) " z )  =  ran  ( rec ( F ,  I
)  |`  z )
37 df-ima 4866 . . . . . . . . . . 11  |-  ( rec ( F ,  (/) ) " z )  =  ran  ( rec ( F ,  (/) )  |`  z )
3835, 36, 373eqtr4g 2488 . . . . . . . . . 10  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  -> 
( rec ( F ,  I ) "
z )  =  ( rec ( F ,  (/) ) " z ) )
3938unieqd 4229 . . . . . . . . 9  |-  ( ( rec ( F ,  I )  |`  z
)  =  ( rec ( F ,  (/) )  |`  z )  ->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) )
40 vex 3083 . . . . . . . . . 10  |-  z  e. 
_V
41 rdglim 7155 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  I
) `  z )  =  U. ( rec ( F ,  I ) " z ) )
42 rdglim 7155 . . . . . . . . . . 11  |-  ( ( z  e.  _V  /\  Lim  z )  ->  ( rec ( F ,  (/) ) `  z )  =  U. ( rec ( F ,  (/) ) "
z ) )
4341, 42eqeq12d 2444 . . . . . . . . . 10  |-  ( ( z  e.  _V  /\  Lim  z )  ->  (
( rec ( F ,  I ) `  z )  =  ( rec ( F ,  (/) ) `  z )  <->  U. ( rec ( F ,  I ) "
z )  =  U. ( rec ( F ,  (/) ) " z ) ) )
4440, 43mpan 674 . . . . . . . . 9  |-  ( Lim  z  ->  ( ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  (/) ) `  z )  <->  U. ( rec ( F ,  I
) " z )  =  U. ( rec ( F ,  (/) ) " z ) ) )
4539, 44syl5ibr 224 . . . . . . . 8  |-  ( Lim  z  ->  ( ( rec ( F ,  I
)  |`  z )  =  ( rec ( F ,  (/) )  |`  z
)  ->  ( rec ( F ,  I ) `
 z )  =  ( rec ( F ,  (/) ) `  z
) ) )
4634, 45sylbird 238 . . . . . . 7  |-  ( Lim  z  ->  ( A. y  e.  z  ( rec ( F ,  I
) `  y )  =  ( rec ( F ,  (/) ) `  y )  ->  ( rec ( F ,  I
) `  z )  =  ( rec ( F ,  (/) ) `  z ) ) )
4746imim2d 54 . . . . . 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 220 . . . . 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 6700 . . . 4  |-  ( x  e.  On  ->  ( -.  I  e.  _V  ->  ( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) ) )
5049com12 32 . . 3  |-  ( -.  I  e.  _V  ->  ( x  e.  On  ->  ( rec ( F ,  I ) `  x
)  =  ( rec ( F ,  (/) ) `  x )
) )
5150ralrimiv 2834 . 2  |-  ( -.  I  e.  _V  ->  A. x  e.  On  ( rec ( F ,  I
) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
52 eqfnfv 5991 . . 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 676 . 2  |-  ( rec ( F ,  I
)  =  rec ( F ,  (/) )  <->  A. x  e.  On  ( rec ( F ,  I ) `  x )  =  ( rec ( F ,  (/) ) `  x ) )
5451, 53sylibr 215 1  |-  ( -.  I  e.  _V  ->  rec ( F ,  I
)  =  rec ( F ,  (/) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   A.wral 2771   _Vcvv 3080    C_ wss 3436   (/)c0 3761   U.cuni 4219   ran crn 4854    |` cres 4855   "cima 4856   Ord word 5441   Oncon0 5442   Lim wlim 5443   suc csuc 5444    Fn wfn 5596   ` cfv 5601   reccrdg 7138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-wrecs 7039  df-recs 7101  df-rdg 7139
This theorem is referenced by:  dfrdg3  30450
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