Theorem frec0g 5922

Description: The initial value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 7-May-2020.)

Assertion

Ref	Expression
frec0g	|- (A e. V -> (frec( F , A) ` (/)) = A)

Proof of Theorem frec0g

Dummy variables g m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dm0 4492	. . . . . . . . . 10 |- dom (/) = (/)
2	1	biantrur 287	. . . . . . . . 9 |- (x e. A <-> ( dom (/) = (/) /\ x e. A))
3		vex 2554	. . . . . . . . . . . . . . . 16 |- m e. _V
4		nsuceq0g 4121	. . . . . . . . . . . . . . . 16 |- ( m e. _V -> suc m =/= (/))
5	3, 4	ax-mp 7	. . . . . . . . . . . . . . 15 |- suc m =/= (/)
6	5	nesymi 2245	. . . . . . . . . . . . . 14 |- -. (/) = suc m
7	1	eqeq1i 2044	. . . . . . . . . . . . . 14 |- ( dom (/) = suc m <-> (/) = suc m)
8	6, 7	mtbir 595	. . . . . . . . . . . . 13 |- -. dom (/) = suc m
9	8	intnanr 838	. . . . . . . . . . . 12 |- -. ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m)))
10	9	a1i 9	. . . . . . . . . . 11 |- ( m e. om -> -. ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))))
11	10	nrex 2405	. . . . . . . . . 10 |- -. E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m)))
12	11	biorfi 664	. . . . . . . . 9 |- (( dom (/) = (/) /\ x e. A) <-> (( dom (/) = (/) /\ x e. A) \/ E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m)))))
13		orcom 646	. . . . . . . . 9 |- ((( dom (/) = (/) /\ x e. A) \/ E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m)))) <-> (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)))
14	2, 12, 13	3bitri 195	. . . . . . . 8 |- (x e. A <-> (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)))
15	14	abbii 2150	. . . . . . 7 |- {x | x e. A } = {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) }
16		abid2 2155	. . . . . . 7 |- {x | x e. A } = A
17	15, 16	eqtr3i 2059	. . . . . 6 |- {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) } = A
18		elex 2560	. . . . . 6 |- (A e. V -> A e. _V)
19	17, 18	syl5eqel 2121	. . . . 5 |- (A e. V -> {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) } e. _V)
20		0ex 3875	. . . . . . 7 |- (/) e. _V
21		dmeq 4478	. . . . . . . . . . . . 13 |- (g = (/) -> dom g = dom (/))
22	21	eqeq1d 2045	. . . . . . . . . . . 12 |- (g = (/) -> ( dom g = suc m <-> dom (/) = suc m))
23		fveq1 5120	. . . . . . . . . . . . . 14 |- (g = (/) -> (g ` m) = ( (/) ` m))
24	23	fveq2d 5125	. . . . . . . . . . . . 13 |- (g = (/) -> ( F ` (g ` m)) = ( F ` ( (/) ` m)))
25	24	eleq2d 2104	. . . . . . . . . . . 12 |- (g = (/) -> (x e. ( F ` (g ` m)) <-> x e. ( F ` ( (/) ` m))))
26	22, 25	anbi12d 442	. . . . . . . . . . 11 |- (g = (/) -> (( dom g = suc m /\ x e. ( F ` (g ` m))) <-> ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m)))))
27	26	rexbidv 2321	. . . . . . . . . 10 |- (g = (/) -> (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) <-> E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m)))))
28	21	eqeq1d 2045	. . . . . . . . . . 11 |- (g = (/) -> ( dom g = (/) <-> dom (/) = (/)))
29	28	anbi1d 438	. . . . . . . . . 10 |- (g = (/) -> (( dom g = (/) /\ x e. A) <-> ( dom (/) = (/) /\ x e. A)))
30	27, 29	orbi12d 706	. . . . . . . . 9 |- (g = (/) -> ((E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) <-> (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A))))
31	30	abbidv 2152	. . . . . . . 8 |- (g = (/) -> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) } = {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) })
32		eqid 2037	. . . . . . . 8 |- (g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) = (g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })
33	31, 32	fvmptg 5191	. . . . . . 7 |- (( (/) e. _V /\ {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) } e. _V) -> ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)) = {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) })
34	20, 33	mpan 400	. . . . . 6 |- ( {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) } e. _V -> ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)) = {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) })
35	34, 17	syl6eq 2085	. . . . 5 |- ( {x | (E. m e. om ( dom (/) = suc m /\ x e. ( F ` ( (/) ` m))) \/ ( dom (/) = (/) /\ x e. A)) } e. _V -> ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)) = A)
36	19, 35	syl 14	. . . 4 |- (A e. V -> ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)) = A)
37	36, 18	eqeltrd 2111	. . 3 |- (A e. V -> ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)) e. _V)
38		df-frec 5918	. . . . . 6 |- frec( F , A) = (recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) |` om)
39	38	fveq1i 5122	. . . . 5 |- (frec( F , A) ` (/)) = ((recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) |` om) ` (/))
40		peano1 4260	. . . . . 6 |- (/) e. om
41		fvres 5141	. . . . . 6 |- ( (/) e. om -> ((recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) |` om) ` (/)) = (recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) ` (/)))
42	40, 41	ax-mp 7	. . . . 5 |- ((recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) |` om) ` (/)) = (recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) ` (/))
43	39, 42	eqtri 2057	. . . 4 |- (frec( F , A) ` (/)) = (recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) ` (/))
44		eqid 2037	. . . . 5 |- recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) = recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }))
45	44	tfr0 5878	. . . 4 |- (((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)) e. _V -> (recs((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) })) ` (/)) = ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)))
46	43, 45	syl5eq 2081	. . 3 |- (((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)) e. _V -> (frec( F , A) ` (/)) = ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)))
47	37, 46	syl 14	. 2 |- (A e. V -> (frec( F , A) ` (/)) = ((g e. _V |-> {x | (E. m e. om ( dom g = suc m /\ x e. ( F ` (g ` m))) \/ ( dom g = (/) /\ x e. A)) }) ` (/)))
48	47, 36	eqtrd 2069	1 |- (A e. V -> (frec( F , A) ` (/)) = A)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97   \/ wo 628   = wceq 1242   e. wcel 1390   {cab 2023   =/= wne 2201  E.wrex 2301   _Vcvv 2551   (/)c0 3218   |-> cmpt 3809   suc csuc 4068   omcom 4256   dom cdm 4288   |` cres 4290   ` cfv 4845  recscrecs 5860  freccfrec 5917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-recs 5861  df-frec 5918

This theorem is referenced by:  frecrdg  5931  freccl  5932  frec2uz0d  8866  frec2uzrdg  8876  frecuzrdg0  8881