Theorem rdgprc0 31699

Description: The value of the recursive definition generator at (/) when the base value is a proper class. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
rdgprc0	|- ( -. I e. _V -> ( rec ( F , I ) ` (/) ) = (/) )

Proof of Theorem rdgprc0

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elon 5778	. . . 4 |- (/) e. On
2		rdgval 7516	. . . 4 |- ( (/) e. On -> ( rec ( F , I ) ` (/) ) = ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ` ( rec ( F , I ) |` (/) ) ) )
3	1, 2	ax-mp 5	. . 3 |- ( rec ( F , I ) ` (/) ) = ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ` ( rec ( F , I ) |` (/) ) )
4		res0 5400	. . . 4 |- ( rec ( F , I ) |` (/) ) = (/)
5	4	fveq2i 6194	. . 3 |- ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ` ( rec ( F , I ) |` (/) ) ) = ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ` (/) )
6	3, 5	eqtri 2644	. 2 |- ( rec ( F , I ) ` (/) ) = ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ` (/) )
7		eqeq1 2626	. . . . . . . 8 |- ( g = (/) -> ( g = (/) <-> (/) = (/) ) )
8		dmeq 5324	. . . . . . . . . 10 |- ( g = (/) -> dom g = dom (/) )
9		limeq 5735	. . . . . . . . . 10 |- ( dom g = dom (/) -> ( Lim dom g <-> Lim dom (/) ) )
10	8, 9	syl 17	. . . . . . . . 9 |- ( g = (/) -> ( Lim dom g <-> Lim dom (/) ) )
11		rneq 5351	. . . . . . . . . 10 |- ( g = (/) -> ran g = ran (/) )
12	11	unieqd 4446	. . . . . . . . 9 |- ( g = (/) -> U. ran g = U. ran (/) )
13		id 22	. . . . . . . . . . 11 |- ( g = (/) -> g = (/) )
14	8	unieqd 4446	. . . . . . . . . . 11 |- ( g = (/) -> U. dom g = U. dom (/) )
15	13, 14	fveq12d 6197	. . . . . . . . . 10 |- ( g = (/) -> ( g ` U. dom g ) = ( (/) ` U. dom (/) ) )
16	15	fveq2d 6195	. . . . . . . . 9 |- ( g = (/) -> ( F ` ( g ` U. dom g ) ) = ( F ` ( (/) ` U. dom (/) ) ) )
17	10, 12, 16	ifbieq12d 4113	. . . . . . . 8 |- ( g = (/) -> if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) = if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) )
18	7, 17	ifbieq2d 4111	. . . . . . 7 |- ( g = (/) -> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) = if ( (/) = (/) , I , if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) ) )
19	18	eleq1d 2686	. . . . . 6 |- ( g = (/) -> ( if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) e. _V <-> if ( (/) = (/) , I , if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) ) e. _V ) )
20		eqid 2622	. . . . . . 7 |- ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) = ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) )
21	20	dmmpt 5630	. . . . . 6 |- dom ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) = { g e. _V | if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) e. _V }
22	19, 21	elrab2 3366	. . . . 5 |- ( (/) e. dom ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) <-> ( (/) e. _V /\ if ( (/) = (/) , I , if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) ) e. _V ) )
23		eqid 2622	. . . . . . . . 9 |- (/) = (/)
24	23	iftruei 4093	. . . . . . . 8 |- if ( (/) = (/) , I , if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) ) = I
25	24	eleq1i 2692	. . . . . . 7 |- ( if ( (/) = (/) , I , if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) ) e. _V <-> I e. _V )
26	25	biimpi 206	. . . . . 6 |- ( if ( (/) = (/) , I , if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) ) e. _V -> I e. _V )
27	26	adantl 482	. . . . 5 |- ( ( (/) e. _V /\ if ( (/) = (/) , I , if ( Lim dom (/) , U. ran (/) , ( F ` ( (/) ` U. dom (/) ) ) ) ) e. _V ) -> I e. _V )
28	22, 27	sylbi 207	. . . 4 |- ( (/) e. dom ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) -> I e. _V )
29	28	con3i 150	. . 3 |- ( -. I e. _V -> -. (/) e. dom ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) )
30		ndmfv 6218	. . 3 |- ( -. (/) e. dom ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) -> ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ` (/) ) = (/) )
31	29, 30	syl 17	. 2 |- ( -. I e. _V -> ( ( g e. _V |-> if ( g = (/) , I , if ( Lim dom g , U. ran g , ( F ` ( g ` U. dom g ) ) ) ) ) ` (/) ) = (/) )
32	6, 31	syl5eq 2668	1 |- ( -. I e. _V -> ( rec ( F , I ) ` (/) ) = (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ifcif 4086   U.cuni 4436   |-> cmpt 4729   dom cdm 5114   ran crn 5115   |` cres 5116   Oncon0 5723   Lim wlim 5724   ` 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-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  rdgprc  31700