Theorem rdgprc 31700

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . . 7 |- ( z = (/) -> ( rec ( F , I ) ` z ) = ( rec ( F , I ) ` (/) ) )
2		fveq2 6191	. . . . . . 7 |- ( z = (/) -> ( rec ( F , (/) ) ` z ) = ( rec ( F , (/) ) ` (/) ) )
3	1, 2	eqeq12d 2637	. . . . . 6 |- ( z = (/) -> ( ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) <-> ( rec ( F , I ) ` (/) ) = ( rec ( F , (/) ) ` (/) ) ) )
4	3	imbi2d 330	. . . . 5 |- ( z = (/) -> ( ( -. I e. _V -> ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) ) <-> ( -. I e. _V -> ( rec ( F , I ) ` (/) ) = ( rec ( F , (/) ) ` (/) ) ) ) )
5		fveq2 6191	. . . . . . 7 |- ( z = y -> ( rec ( F , I ) ` z ) = ( rec ( F , I ) ` y ) )
6		fveq2 6191	. . . . . . 7 |- ( z = y -> ( rec ( F , (/) ) ` z ) = ( rec ( F , (/) ) ` y ) )
7	5, 6	eqeq12d 2637	. . . . . 6 |- ( z = y -> ( ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) <-> ( rec ( F , I ) ` y ) = ( rec ( F , (/) ) ` y ) ) )
8	7	imbi2d 330	. . . . 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 6191	. . . . . . 7 |- ( z = suc y -> ( rec ( F , I ) ` z ) = ( rec ( F , I ) ` suc y ) )
10		fveq2 6191	. . . . . . 7 |- ( z = suc y -> ( rec ( F , (/) ) ` z ) = ( rec ( F , (/) ) ` suc y ) )
11	9, 10	eqeq12d 2637	. . . . . 6 |- ( z = suc y -> ( ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) <-> ( rec ( F , I ) ` suc y ) = ( rec ( F , (/) ) ` suc y ) ) )
12	11	imbi2d 330	. . . . 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 6191	. . . . . . 7 |- ( z = x -> ( rec ( F , I ) ` z ) = ( rec ( F , I ) ` x ) )
14		fveq2 6191	. . . . . . 7 |- ( z = x -> ( rec ( F , (/) ) ` z ) = ( rec ( F , (/) ) ` x ) )
15	13, 14	eqeq12d 2637	. . . . . 6 |- ( z = x -> ( ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) <-> ( rec ( F , I ) ` x ) = ( rec ( F , (/) ) ` x ) ) )
16	15	imbi2d 330	. . . . 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 31699	. . . . . 6 |- ( -. I e. _V -> ( rec ( F , I ) ` (/) ) = (/) )
18		0ex 4790	. . . . . . 7 |- (/) e. _V
19	18	rdg0 7517	. . . . . 6 |- ( rec ( F , (/) ) ` (/) ) = (/)
20	17, 19	syl6eqr 2674	. . . . 5 |- ( -. I e. _V -> ( rec ( F , I ) ` (/) ) = ( rec ( F , (/) ) ` (/) ) )
21		fveq2 6191	. . . . . . 7 |- ( ( rec ( F , I ) ` y ) = ( rec ( F , (/) ) ` y ) -> ( F ` ( rec ( F , I ) ` y ) ) = ( F ` ( rec ( F , (/) ) ` y ) ) )
22		rdgsuc 7520	. . . . . . . 8 |- ( y e. On -> ( rec ( F , I ) ` suc y ) = ( F ` ( rec ( F , I ) ` y ) ) )
23		rdgsuc 7520	. . . . . . . 8 |- ( y e. On -> ( rec ( F , (/) ) ` suc y ) = ( F ` ( rec ( F , (/) ) ` y ) ) )
24	22, 23	eqeq12d 2637	. . . . . . 7 |- ( y e. On -> ( ( rec ( F , I ) ` suc y ) = ( rec ( F , (/) ) ` suc y ) <-> ( F ` ( rec ( F , I ) ` y ) ) = ( F ` ( rec ( F , (/) ) ` y ) ) ) )
25	21, 24	syl5ibr 236	. . . . . 6 |- ( y e. On -> ( ( rec ( F , I ) ` y ) = ( rec ( F , (/) ) ` y ) -> ( rec ( F , I ) ` suc y ) = ( rec ( F , (/) ) ` suc y ) ) )
26	25	imim2d 57	. . . . 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 2960	. . . . . 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 5784	. . . . . . . . 9 |- ( Lim z -> Ord z )
29		ordsson 6989	. . . . . . . . 9 |- ( Ord z -> z C_ On )
30		rdgfnon 7514	. . . . . . . . . 10 |- rec ( F , I ) Fn On
31		rdgfnon 7514	. . . . . . . . . 10 |- rec ( F , (/) ) Fn On
32		fvreseq 6319	. . . . . . . . . 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 ) ) )
33	30, 31, 32	mpanl12 718	. . . . . . . . 9 |- ( z C_ On -> ( ( rec ( F , I ) |` z ) = ( rec ( F , (/) ) |` z ) <-> A. y e. z ( rec ( F , I ) ` y ) = ( rec ( F , (/) ) ` y ) ) )
34	28, 29, 33	3syl 18	. . . . . . . 8 |- ( Lim z -> ( ( rec ( F , I ) |` z ) = ( rec ( F , (/) ) |` z ) <-> A. y e. z ( rec ( F , I ) ` y ) = ( rec ( F , (/) ) ` y ) ) )
35		rneq 5351	. . . . . . . . . . 11 |- ( ( rec ( F , I ) |` z ) = ( rec ( F , (/) ) |` z ) -> ran ( rec ( F , I ) |` z ) = ran ( rec ( F , (/) ) |` z ) )
36		df-ima 5127	. . . . . . . . . . 11 |- ( rec ( F , I ) " z ) = ran ( rec ( F , I ) |` z )
37		df-ima 5127	. . . . . . . . . . 11 |- ( rec ( F , (/) ) " z ) = ran ( rec ( F , (/) ) |` z )
38	35, 36, 37	3eqtr4g 2681	. . . . . . . . . 10 |- ( ( rec ( F , I ) |` z ) = ( rec ( F , (/) ) |` z ) -> ( rec ( F , I ) " z ) = ( rec ( F , (/) ) " z ) )
39	38	unieqd 4446	. . . . . . . . 9 |- ( ( rec ( F , I ) |` z ) = ( rec ( F , (/) ) |` z ) -> U. ( rec ( F , I ) " z ) = U. ( rec ( F , (/) ) " z ) )
40		vex 3203	. . . . . . . . . 10 |- z e. _V
41		rdglim 7522	. . . . . . . . . . 11 |- ( ( z e. _V /\ Lim z ) -> ( rec ( F , I ) ` z ) = U. ( rec ( F , I ) " z ) )
42		rdglim 7522	. . . . . . . . . . 11 |- ( ( z e. _V /\ Lim z ) -> ( rec ( F , (/) ) ` z ) = U. ( rec ( F , (/) ) " z ) )
43	41, 42	eqeq12d 2637	. . . . . . . . . 10 |- ( ( z e. _V /\ Lim z ) -> ( ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) <-> U. ( rec ( F , I ) " z ) = U. ( rec ( F , (/) ) " z ) ) )
44	40, 43	mpan 706	. . . . . . . . 9 |- ( Lim z -> ( ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) <-> U. ( rec ( F , I ) " z ) = U. ( rec ( F , (/) ) " z ) ) )
45	39, 44	syl5ibr 236	. . . . . . . 8 |- ( Lim z -> ( ( rec ( F , I ) |` z ) = ( rec ( F , (/) ) |` z ) -> ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) ) )
46	34, 45	sylbird 250	. . . . . . 7 |- ( Lim z -> ( A. y e. z ( rec ( F , I ) ` y ) = ( rec ( F , (/) ) ` y ) -> ( rec ( F , I ) ` z ) = ( rec ( F , (/) ) ` z ) ) )
47	46	imim2d 57	. . . . . 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 ) ) ) )
48	27, 47	syl5bi 232	. . . . 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 ) ) ) )
49	4, 8, 12, 16, 20, 26, 48	tfinds 7059	. . . 4 |- ( x e. On -> ( -. I e. _V -> ( rec ( F , I ) ` x ) = ( rec ( F , (/) ) ` x ) ) )
50	49	com12 32	. . 3 |- ( -. I e. _V -> ( x e. On -> ( rec ( F , I ) ` x ) = ( rec ( F , (/) ) ` x ) ) )
51	50	ralrimiv 2965	. 2 |- ( -. I e. _V -> A. x e. On ( rec ( F , I ) ` x ) = ( rec ( F , (/) ) ` x ) )
52		eqfnfv 6311	. . 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 ) ) )
53	30, 31, 52	mp2an 708	. 2 |- ( rec ( F , I ) = rec ( F , (/) ) <-> A. x e. On ( rec ( F , I ) ` x ) = ( rec ( F , (/) ) ` x ) )
54	51, 53	sylibr 224	1 |- ( -. I e. _V -> rec ( F , I ) = rec ( F , (/) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   (/)c0 3915   U.cuni 4436   ran crn 5115   |` cres 5116   "cima 5117   Ord word 5722   Oncon0 5723   Lim wlim 5724   suc csuc 5725   Fn wfn 5883   ` cfv 5888   reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  dfrdg3  31702