Theorem frecxp 6314

Description: Subset relationship for the finite recursive function generator. (Contributed by Scott Fenton, 30-Jul-2019.)

Hypotheses

Ref	Expression
frecxp.1	|- F = FRec (G, I)
frecxp.2	|- G e. _V

Assertion

Ref	Expression
frecxp	|- F C_ ( Nn X. (ran G u. {I}))

Proof of Theorem frecxp

Dummy variables x y z a b c d i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecxp.1	. 2 |- F = FRec (G, I)
2		eqid 2353	. . . . . 6 |- G = G
3		freceq12 6311	. . . . . 6 |- ((G = G /\ i = I) -> FRec (G, i) = FRec (G, I))
4	2, 3	mpan 651	. . . . 5 |- (i = I -> FRec (G, i) = FRec (G, I))
5		sneq 3744	. . . . . . 7 |- (i = I -> {i} = {I})
6	5	uneq2d 3418	. . . . . 6 |- (i = I -> (ran G u. {i}) = (ran G u. {I}))
7	6	xpeq2d 4808	. . . . 5 |- (i = I -> ( Nn X. (ran G u. {i})) = ( Nn X. (ran G u. {I})))
8	4, 7	sseq12d 3300	. . . 4 |- (i = I -> ( FRec (G, i) C_ ( Nn X. (ran G u. {i})) <-> FRec (G, I) C_ ( Nn X. (ran G u. {I}))))
9		nncex 4396	. . . . . 6 |- Nn e. _V
10		frecxp.2	. . . . . . . 8 |- G e. _V
11	10	rnex 5107	. . . . . . 7 |- ran G e. _V
12		snex 4111	. . . . . . 7 |- {i} e. _V
13	11, 12	unex 4106	. . . . . 6 |- (ran G u. {i}) e. _V
14	9, 13	xpex 5115	. . . . 5 |- ( Nn X. (ran G u. {i})) e. _V
15		peano1 4402	. . . . . 6 |- 0c e. Nn
16		vex 2862	. . . . . . . 8 |- i e. _V
17	16	snid 3760	. . . . . . 7 |- i e. {i}
18		elun2 3431	. . . . . . 7 |- (i e. {i} -> i e. (ran G u. {i}))
19	17, 18	ax-mp 5	. . . . . 6 |- i e. (ran G u. {i})
20		0cex 4392	. . . . . . . . 9 |- 0c e. _V
21	20, 16	opex 4588	. . . . . . . 8 |- <.0c, i>. e. _V
22	21	snss 3838	. . . . . . 7 |- (<.0c, i>. e. ( Nn X. (ran G u. {i})) <-> {<.0c, i>.} C_ ( Nn X. (ran G u. {i})))
23		opelxp 4811	. . . . . . 7 |- (<.0c, i>. e. ( Nn X. (ran G u. {i})) <-> (0c e. Nn /\ i e. (ran G u. {i})))
24	22, 23	bitr3i 242	. . . . . 6 |- ({<.0c, i>.} C_ ( Nn X. (ran G u. {i})) <-> (0c e. Nn /\ i e. (ran G u. {i})))
25	15, 19, 24	mpbir2an 886	. . . . 5 |- {<.0c, i>.} C_ ( Nn X. (ran G u. {i}))
26		brpprod 5839	. . . . . . . . 9 |- (y PProd ((x e. _V |-> (x +o 1c)), G)z <-> E.aE.bE.cE.d(y = <.a, b>. /\ z = <.c, d>. /\ (a(x e. _V |-> (x +o 1c))c /\ bGd)))
27		vex 2862	. . . . . . . . . . . . . . . 16 |- a e. _V
28		vex 2862	. . . . . . . . . . . . . . . 16 |- c e. _V
29	27, 28	brcsuc 6260	. . . . . . . . . . . . . . 15 |- (a(x e. _V |-> (x +o 1c))c <-> c = (a +o 1c))
30		brelrn 4960	. . . . . . . . . . . . . . . . . . . 20 |- (bGd -> d e. ran G)
31		elun1 3430	. . . . . . . . . . . . . . . . . . . 20 |- (d e. ran G -> d e. (ran G u. {i}))
32	30, 31	syl 15	. . . . . . . . . . . . . . . . . . 19 |- (bGd -> d e. (ran G u. {i}))
33		peano2 4403	. . . . . . . . . . . . . . . . . . 19 |- (a e. Nn -> (a +o 1c) e. Nn )
34	32, 33	anim12ci 550	. . . . . . . . . . . . . . . . . 18 |- ((bGd /\ a e. Nn ) -> ((a +o 1c) e. Nn /\ d e. (ran G u. {i})))
35	34	adantrr 697	. . . . . . . . . . . . . . . . 17 |- ((bGd /\ (a e. Nn /\ b e. (ran G u. {i}))) -> ((a +o 1c) e. Nn /\ d e. (ran G u. {i})))
36		eleq1 2413	. . . . . . . . . . . . . . . . . 18 |- (c = (a +o 1c) -> (c e. Nn <-> (a +o 1c) e. Nn ))
37	36	anbi1d 685	. . . . . . . . . . . . . . . . 17 |- (c = (a +o 1c) -> ((c e. Nn /\ d e. (ran G u. {i})) <-> ((a +o 1c) e. Nn /\ d e. (ran G u. {i}))))
38	35, 37	syl5ibr 212	. . . . . . . . . . . . . . . 16 |- (c = (a +o 1c) -> ((bGd /\ (a e. Nn /\ b e. (ran G u. {i}))) -> (c e. Nn /\ d e. (ran G u. {i}))))
39	38	exp3a 425	. . . . . . . . . . . . . . 15 |- (c = (a +o 1c) -> (bGd -> ((a e. Nn /\ b e. (ran G u. {i})) -> (c e. Nn /\ d e. (ran G u. {i})))))
40	29, 39	sylbi 187	. . . . . . . . . . . . . 14 |- (a(x e. _V |-> (x +o 1c))c -> (bGd -> ((a e. Nn /\ b e. (ran G u. {i})) -> (c e. Nn /\ d e. (ran G u. {i})))))
41	40	imp 418	. . . . . . . . . . . . 13 |- ((a(x e. _V |-> (x +o 1c))c /\ bGd) -> ((a e. Nn /\ b e. (ran G u. {i})) -> (c e. Nn /\ d e. (ran G u. {i}))))
42		eleq1 2413	. . . . . . . . . . . . . . . 16 |- (y = <.a, b>. -> (y e. ( Nn X. (ran G u. {i})) <-> <.a, b>. e. ( Nn X. (ran G u. {i}))))
43		opelxp 4811	. . . . . . . . . . . . . . . 16 |- (<.a, b>. e. ( Nn X. (ran G u. {i})) <-> (a e. Nn /\ b e. (ran G u. {i})))
44	42, 43	syl6bb 252	. . . . . . . . . . . . . . 15 |- (y = <.a, b>. -> (y e. ( Nn X. (ran G u. {i})) <-> (a e. Nn /\ b e. (ran G u. {i}))))
45	44	adantr 451	. . . . . . . . . . . . . 14 |- ((y = <.a, b>. /\ z = <.c, d>.) -> (y e. ( Nn X. (ran G u. {i})) <-> (a e. Nn /\ b e. (ran G u. {i}))))
46		eleq1 2413	. . . . . . . . . . . . . . . 16 |- (z = <.c, d>. -> (z e. ( Nn X. (ran G u. {i})) <-> <.c, d>. e. ( Nn X. (ran G u. {i}))))
47		opelxp 4811	. . . . . . . . . . . . . . . 16 |- (<.c, d>. e. ( Nn X. (ran G u. {i})) <-> (c e. Nn /\ d e. (ran G u. {i})))
48	46, 47	syl6bb 252	. . . . . . . . . . . . . . 15 |- (z = <.c, d>. -> (z e. ( Nn X. (ran G u. {i})) <-> (c e. Nn /\ d e. (ran G u. {i}))))
49	48	adantl 452	. . . . . . . . . . . . . 14 |- ((y = <.a, b>. /\ z = <.c, d>.) -> (z e. ( Nn X. (ran G u. {i})) <-> (c e. Nn /\ d e. (ran G u. {i}))))
50	45, 49	imbi12d 311	. . . . . . . . . . . . 13 |- ((y = <.a, b>. /\ z = <.c, d>.) -> ((y e. ( Nn X. (ran G u. {i})) -> z e. ( Nn X. (ran G u. {i}))) <-> ((a e. Nn /\ b e. (ran G u. {i})) -> (c e. Nn /\ d e. (ran G u. {i})))))
51	41, 50	syl5ibr 212	. . . . . . . . . . . 12 |- ((y = <.a, b>. /\ z = <.c, d>.) -> ((a(x e. _V |-> (x +o 1c))c /\ bGd) -> (y e. ( Nn X. (ran G u. {i})) -> z e. ( Nn X. (ran G u. {i})))))
52	51	3impia 1148	. . . . . . . . . . 11 |- ((y = <.a, b>. /\ z = <.c, d>. /\ (a(x e. _V |-> (x +o 1c))c /\ bGd)) -> (y e. ( Nn X. (ran G u. {i})) -> z e. ( Nn X. (ran G u. {i}))))
53	52	exlimivv 1635	. . . . . . . . . 10 |- (E.cE.d(y = <.a, b>. /\ z = <.c, d>. /\ (a(x e. _V |-> (x +o 1c))c /\ bGd)) -> (y e. ( Nn X. (ran G u. {i})) -> z e. ( Nn X. (ran G u. {i}))))
54	53	exlimivv 1635	. . . . . . . . 9 |- (E.aE.bE.cE.d(y = <.a, b>. /\ z = <.c, d>. /\ (a(x e. _V |-> (x +o 1c))c /\ bGd)) -> (y e. ( Nn X. (ran G u. {i})) -> z e. ( Nn X. (ran G u. {i}))))
55	26, 54	sylbi 187	. . . . . . . 8 |- (y PProd ((x e. _V |-> (x +o 1c)), G)z -> (y e. ( Nn X. (ran G u. {i})) -> z e. ( Nn X. (ran G u. {i}))))
56	55	impcom 419	. . . . . . 7 |- ((y e. ( Nn X. (ran G u. {i})) /\ y PProd ((x e. _V |-> (x +o 1c)), G)z) -> z e. ( Nn X. (ran G u. {i})))
57	56	ax-gen 1546	. . . . . 6 |- A.z((y e. ( Nn X. (ran G u. {i})) /\ y PProd ((x e. _V |-> (x +o 1c)), G)z) -> z e. ( Nn X. (ran G u. {i})))
58	57	rgenw 2681	. . . . 5 |- A.y e. FRec (G, i)A.z((y e. ( Nn X. (ran G u. {i})) /\ y PProd ((x e. _V |-> (x +o 1c)), G)z) -> z e. ( Nn X. (ran G u. {i})))
59		snex 4111	. . . . . 6 |- {<.0c, i>.} e. _V
60		csucex 6259	. . . . . . 7 |- (x e. _V |-> (x +o 1c)) e. _V
61	60, 10	pprodex 5838	. . . . . 6 |- PProd ((x e. _V |-> (x +o 1c)), G) e. _V
62		df-frec 6310	. . . . . 6 |- FRec (G, i) = Clos1 ({<.0c, i>.}, PProd ((x e. _V |-> (x +o 1c)), G))
63	59, 61, 62	clos1induct 5880	. . . . 5 |- ((( Nn X. (ran G u. {i})) e. _V /\ {<.0c, i>.} C_ ( Nn X. (ran G u. {i})) /\ A.y e. FRec (G, i)A.z((y e. ( Nn X. (ran G u. {i})) /\ y PProd ((x e. _V |-> (x +o 1c)), G)z) -> z e. ( Nn X. (ran G u. {i})))) -> FRec (G, i) C_ ( Nn X. (ran G u. {i})))
64	14, 25, 58, 63	mp3an 1277	. . . 4 |- FRec (G, i) C_ ( Nn X. (ran G u. {i}))
65	8, 64	vtoclg 2914	. . 3 |- (I e. _V -> FRec (G, I) C_ ( Nn X. (ran G u. {I})))
66		df-frec 6310	. . . 4 |- FRec (G, I) = Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), G))
67		opexb 4603	. . . . . . . . . 10 |- (<.0c, I>. e. _V <-> (0c e. _V /\ I e. _V))
68	67	simprbi 450	. . . . . . . . 9 |- (<.0c, I>. e. _V -> I e. _V)
69	68	con3i 127	. . . . . . . 8 |- (-. I e. _V -> -. <.0c, I>. e. _V)
70		snprc 3788	. . . . . . . 8 |- (-. <.0c, I>. e. _V <-> {<.0c, I>.} = (/))
71	69, 70	sylib 188	. . . . . . 7 |- (-. I e. _V -> {<.0c, I>.} = (/))
72		clos1eq1 5874	. . . . . . 7 |- ({<.0c, I>.} = (/) -> Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), G)) = Clos1 ((/), PProd ((x e. _V |-> (x +o 1c)), G)))
73	71, 72	syl 15	. . . . . 6 |- (-. I e. _V -> Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), G)) = Clos1 ((/), PProd ((x e. _V |-> (x +o 1c)), G)))
74		eqid 2353	. . . . . . 7 |- Clos1 ((/), PProd ((x e. _V |-> (x +o 1c)), G)) = Clos1 ((/), PProd ((x e. _V |-> (x +o 1c)), G))
75	61, 74	clos10 5887	. . . . . 6 |- Clos1 ((/), PProd ((x e. _V |-> (x +o 1c)), G)) = (/)
76	73, 75	syl6eq 2401	. . . . 5 |- (-. I e. _V -> Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), G)) = (/))
77		0ss 3579	. . . . 5 |- (/) C_ ( Nn X. (ran G u. {I}))
78	76, 77	syl6eqss 3321	. . . 4 |- (-. I e. _V -> Clos1 ({<.0c, I>.}, PProd ((x e. _V |-> (x +o 1c)), G)) C_ ( Nn X. (ran G u. {I})))
79	66, 78	syl5eqss 3315	. . 3 |- (-. I e. _V -> FRec (G, I) C_ ( Nn X. (ran G u. {I})))
80	65, 79	pm2.61i 156	. 2 |- FRec (G, I) C_ ( Nn X. (ran G u. {I}))
81	1, 80	eqsstri 3301	1 |- F C_ ( Nn X. (ran G u. {I}))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  A.wral 2614  _Vcvv 2859   u. cun 3207   C_ wss 3257  (/)c0 3550  {csn 3737  1_cc1c 4134   Nn cnnc 4373  0_cc0c 4374   +o cplc 4375  <.cop 4561   class class class wbr 4639   X. cxp 4770  ran crn 4773   |-> cmpt 5651   PProd cpprod 5737   Clos1 cclos1 5872   FRec cfrec 6309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-csb 3137  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-fo 4793  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-pprod 5738  df-fix 5740  df-cup 5742  df-disj 5744  df-addcfn 5746  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-clos1 5873  df-frec 6310

This theorem is referenced by:  frecxpg  6315