Theorem frecxp 6315

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 ∈ V

Assertion

Ref	Expression
frecxp	⊢ F ⊆ ( Nn × (ran G ∪ {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 6312	. . . . . 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 3745	. . . . . . 7 ⊢ (i = I → {i} = {I})
6	5	uneq2d 3419	. . . . . 6 ⊢ (i = I → (ran G ∪ {i}) = (ran G ∪ {I}))
7	6	xpeq2d 4809	. . . . 5 ⊢ (i = I → ( Nn × (ran G ∪ {i})) = ( Nn × (ran G ∪ {I})))
8	4, 7	sseq12d 3301	. . . 4 ⊢ (i = I → ( FRec (G, i) ⊆ ( Nn × (ran G ∪ {i})) ↔ FRec (G, I) ⊆ ( Nn × (ran G ∪ {I}))))
9		nncex 4397	. . . . . 6 ⊢ Nn ∈ V
10		frecxp.2	. . . . . . . 8 ⊢ G ∈ V
11	10	rnex 5108	. . . . . . 7 ⊢ ran G ∈ V
12		snex 4112	. . . . . . 7 ⊢ {i} ∈ V
13	11, 12	unex 4107	. . . . . 6 ⊢ (ran G ∪ {i}) ∈ V
14	9, 13	xpex 5116	. . . . 5 ⊢ ( Nn × (ran G ∪ {i})) ∈ V
15		peano1 4403	. . . . . 6 ⊢ 0c ∈ Nn
16		vex 2863	. . . . . . . 8 ⊢ i ∈ V
17	16	snid 3761	. . . . . . 7 ⊢ i ∈ {i}
18		elun2 3432	. . . . . . 7 ⊢ (i ∈ {i} → i ∈ (ran G ∪ {i}))
19	17, 18	ax-mp 5	. . . . . 6 ⊢ i ∈ (ran G ∪ {i})
20		0cex 4393	. . . . . . . . 9 ⊢ 0c ∈ V
21	20, 16	opex 4589	. . . . . . . 8 ⊢ ⟨0c, i⟩ ∈ V
22	21	snss 3839	. . . . . . 7 ⊢ (⟨0c, i⟩ ∈ ( Nn × (ran G ∪ {i})) ↔ {⟨0c, i⟩} ⊆ ( Nn × (ran G ∪ {i})))
23		opelxp 4812	. . . . . . 7 ⊢ (⟨0c, i⟩ ∈ ( Nn × (ran G ∪ {i})) ↔ (0c ∈ Nn ∧ i ∈ (ran G ∪ {i})))
24	22, 23	bitr3i 242	. . . . . 6 ⊢ ({⟨0c, i⟩} ⊆ ( Nn × (ran G ∪ {i})) ↔ (0c ∈ Nn ∧ i ∈ (ran G ∪ {i})))
25	15, 19, 24	mpbir2an 886	. . . . 5 ⊢ {⟨0c, i⟩} ⊆ ( Nn × (ran G ∪ {i}))
26		brpprod 5840	. . . . . . . . 9 ⊢ (y PProd ((x ∈ V ↦ (x +c 1c)), G)z ↔ ∃a∃b∃c∃d(y = ⟨a, b⟩ ∧ z = ⟨c, d⟩ ∧ (a(x ∈ V ↦ (x +c 1c))c ∧ bGd)))
27		vex 2863	. . . . . . . . . . . . . . . 16 ⊢ a ∈ V
28		vex 2863	. . . . . . . . . . . . . . . 16 ⊢ c ∈ V
29	27, 28	brcsuc 6261	. . . . . . . . . . . . . . 15 ⊢ (a(x ∈ V ↦ (x +c 1c))c ↔ c = (a +c 1c))
30		brelrn 4961	. . . . . . . . . . . . . . . . . . . 20 ⊢ (bGd → d ∈ ran G)
31		elun1 3431	. . . . . . . . . . . . . . . . . . . 20 ⊢ (d ∈ ran G → d ∈ (ran G ∪ {i}))
32	30, 31	syl 15	. . . . . . . . . . . . . . . . . . 19 ⊢ (bGd → d ∈ (ran G ∪ {i}))
33		peano2 4404	. . . . . . . . . . . . . . . . . . 19 ⊢ (a ∈ Nn → (a +c 1c) ∈ Nn )
34	32, 33	anim12ci 550	. . . . . . . . . . . . . . . . . 18 ⊢ ((bGd ∧ a ∈ Nn ) → ((a +c 1c) ∈ Nn ∧ d ∈ (ran G ∪ {i})))
35	34	adantrr 697	. . . . . . . . . . . . . . . . 17 ⊢ ((bGd ∧ (a ∈ Nn ∧ b ∈ (ran G ∪ {i}))) → ((a +c 1c) ∈ Nn ∧ d ∈ (ran G ∪ {i})))
36		eleq1 2413	. . . . . . . . . . . . . . . . . 18 ⊢ (c = (a +c 1c) → (c ∈ Nn ↔ (a +c 1c) ∈ Nn ))
37	36	anbi1d 685	. . . . . . . . . . . . . . . . 17 ⊢ (c = (a +c 1c) → ((c ∈ Nn ∧ d ∈ (ran G ∪ {i})) ↔ ((a +c 1c) ∈ Nn ∧ d ∈ (ran G ∪ {i}))))
38	35, 37	syl5ibr 212	. . . . . . . . . . . . . . . 16 ⊢ (c = (a +c 1c) → ((bGd ∧ (a ∈ Nn ∧ b ∈ (ran G ∪ {i}))) → (c ∈ Nn ∧ d ∈ (ran G ∪ {i}))))
39	38	exp3a 425	. . . . . . . . . . . . . . 15 ⊢ (c = (a +c 1c) → (bGd → ((a ∈ Nn ∧ b ∈ (ran G ∪ {i})) → (c ∈ Nn ∧ d ∈ (ran G ∪ {i})))))
40	29, 39	sylbi 187	. . . . . . . . . . . . . 14 ⊢ (a(x ∈ V ↦ (x +c 1c))c → (bGd → ((a ∈ Nn ∧ b ∈ (ran G ∪ {i})) → (c ∈ Nn ∧ d ∈ (ran G ∪ {i})))))
41	40	imp 418	. . . . . . . . . . . . 13 ⊢ ((a(x ∈ V ↦ (x +c 1c))c ∧ bGd) → ((a ∈ Nn ∧ b ∈ (ran G ∪ {i})) → (c ∈ Nn ∧ d ∈ (ran G ∪ {i}))))
42		eleq1 2413	. . . . . . . . . . . . . . . 16 ⊢ (y = ⟨a, b⟩ → (y ∈ ( Nn × (ran G ∪ {i})) ↔ ⟨a, b⟩ ∈ ( Nn × (ran G ∪ {i}))))
43		opelxp 4812	. . . . . . . . . . . . . . . 16 ⊢ (⟨a, b⟩ ∈ ( Nn × (ran G ∪ {i})) ↔ (a ∈ Nn ∧ b ∈ (ran G ∪ {i})))
44	42, 43	syl6bb 252	. . . . . . . . . . . . . . 15 ⊢ (y = ⟨a, b⟩ → (y ∈ ( Nn × (ran G ∪ {i})) ↔ (a ∈ Nn ∧ b ∈ (ran G ∪ {i}))))
45	44	adantr 451	. . . . . . . . . . . . . 14 ⊢ ((y = ⟨a, b⟩ ∧ z = ⟨c, d⟩) → (y ∈ ( Nn × (ran G ∪ {i})) ↔ (a ∈ Nn ∧ b ∈ (ran G ∪ {i}))))
46		eleq1 2413	. . . . . . . . . . . . . . . 16 ⊢ (z = ⟨c, d⟩ → (z ∈ ( Nn × (ran G ∪ {i})) ↔ ⟨c, d⟩ ∈ ( Nn × (ran G ∪ {i}))))
47		opelxp 4812	. . . . . . . . . . . . . . . 16 ⊢ (⟨c, d⟩ ∈ ( Nn × (ran G ∪ {i})) ↔ (c ∈ Nn ∧ d ∈ (ran G ∪ {i})))
48	46, 47	syl6bb 252	. . . . . . . . . . . . . . 15 ⊢ (z = ⟨c, d⟩ → (z ∈ ( Nn × (ran G ∪ {i})) ↔ (c ∈ Nn ∧ d ∈ (ran G ∪ {i}))))
49	48	adantl 452	. . . . . . . . . . . . . 14 ⊢ ((y = ⟨a, b⟩ ∧ z = ⟨c, d⟩) → (z ∈ ( Nn × (ran G ∪ {i})) ↔ (c ∈ Nn ∧ d ∈ (ran G ∪ {i}))))
50	45, 49	imbi12d 311	. . . . . . . . . . . . 13 ⊢ ((y = ⟨a, b⟩ ∧ z = ⟨c, d⟩) → ((y ∈ ( Nn × (ran G ∪ {i})) → z ∈ ( Nn × (ran G ∪ {i}))) ↔ ((a ∈ Nn ∧ b ∈ (ran G ∪ {i})) → (c ∈ Nn ∧ d ∈ (ran G ∪ {i})))))
51	41, 50	syl5ibr 212	. . . . . . . . . . . 12 ⊢ ((y = ⟨a, b⟩ ∧ z = ⟨c, d⟩) → ((a(x ∈ V ↦ (x +c 1c))c ∧ bGd) → (y ∈ ( Nn × (ran G ∪ {i})) → z ∈ ( Nn × (ran G ∪ {i})))))
52	51	3impia 1148	. . . . . . . . . . 11 ⊢ ((y = ⟨a, b⟩ ∧ z = ⟨c, d⟩ ∧ (a(x ∈ V ↦ (x +c 1c))c ∧ bGd)) → (y ∈ ( Nn × (ran G ∪ {i})) → z ∈ ( Nn × (ran G ∪ {i}))))
53	52	exlimivv 1635	. . . . . . . . . 10 ⊢ (∃c∃d(y = ⟨a, b⟩ ∧ z = ⟨c, d⟩ ∧ (a(x ∈ V ↦ (x +c 1c))c ∧ bGd)) → (y ∈ ( Nn × (ran G ∪ {i})) → z ∈ ( Nn × (ran G ∪ {i}))))
54	53	exlimivv 1635	. . . . . . . . 9 ⊢ (∃a∃b∃c∃d(y = ⟨a, b⟩ ∧ z = ⟨c, d⟩ ∧ (a(x ∈ V ↦ (x +c 1c))c ∧ bGd)) → (y ∈ ( Nn × (ran G ∪ {i})) → z ∈ ( Nn × (ran G ∪ {i}))))
55	26, 54	sylbi 187	. . . . . . . 8 ⊢ (y PProd ((x ∈ V ↦ (x +c 1c)), G)z → (y ∈ ( Nn × (ran G ∪ {i})) → z ∈ ( Nn × (ran G ∪ {i}))))
56	55	impcom 419	. . . . . . 7 ⊢ ((y ∈ ( Nn × (ran G ∪ {i})) ∧ y PProd ((x ∈ V ↦ (x +c 1c)), G)z) → z ∈ ( Nn × (ran G ∪ {i})))
57	56	ax-gen 1546	. . . . . 6 ⊢ ∀z((y ∈ ( Nn × (ran G ∪ {i})) ∧ y PProd ((x ∈ V ↦ (x +c 1c)), G)z) → z ∈ ( Nn × (ran G ∪ {i})))
58	57	rgenw 2682	. . . . 5 ⊢ ∀y ∈ FRec (G, i)∀z((y ∈ ( Nn × (ran G ∪ {i})) ∧ y PProd ((x ∈ V ↦ (x +c 1c)), G)z) → z ∈ ( Nn × (ran G ∪ {i})))
59		snex 4112	. . . . . 6 ⊢ {⟨0c, i⟩} ∈ V
60		csucex 6260	. . . . . . 7 ⊢ (x ∈ V ↦ (x +c 1c)) ∈ V
61	60, 10	pprodex 5839	. . . . . 6 ⊢ PProd ((x ∈ V ↦ (x +c 1c)), G) ∈ V
62		df-frec 6311	. . . . . 6 ⊢ FRec (G, i) = Clos1 ({⟨0c, i⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G))
63	59, 61, 62	clos1induct 5881	. . . . 5 ⊢ ((( Nn × (ran G ∪ {i})) ∈ V ∧ {⟨0c, i⟩} ⊆ ( Nn × (ran G ∪ {i})) ∧ ∀y ∈ FRec (G, i)∀z((y ∈ ( Nn × (ran G ∪ {i})) ∧ y PProd ((x ∈ V ↦ (x +c 1c)), G)z) → z ∈ ( Nn × (ran G ∪ {i})))) → FRec (G, i) ⊆ ( Nn × (ran G ∪ {i})))
64	14, 25, 58, 63	mp3an 1277	. . . 4 ⊢ FRec (G, i) ⊆ ( Nn × (ran G ∪ {i}))
65	8, 64	vtoclg 2915	. . 3 ⊢ (I ∈ V → FRec (G, I) ⊆ ( Nn × (ran G ∪ {I})))
66		df-frec 6311	. . . 4 ⊢ FRec (G, I) = Clos1 ({⟨0c, I⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G))
67		opexb 4604	. . . . . . . . . 10 ⊢ (⟨0c, I⟩ ∈ V ↔ (0c ∈ V ∧ I ∈ V))
68	67	simprbi 450	. . . . . . . . 9 ⊢ (⟨0c, I⟩ ∈ V → I ∈ V)
69	68	con3i 127	. . . . . . . 8 ⊢ (¬ I ∈ V → ¬ ⟨0c, I⟩ ∈ V)
70		snprc 3789	. . . . . . . 8 ⊢ (¬ ⟨0c, I⟩ ∈ V ↔ {⟨0c, I⟩} = ∅)
71	69, 70	sylib 188	. . . . . . 7 ⊢ (¬ I ∈ V → {⟨0c, I⟩} = ∅)
72		clos1eq1 5875	. . . . . . 7 ⊢ ({⟨0c, I⟩} = ∅ → Clos1 ({⟨0c, I⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G)) = Clos1 (∅, PProd ((x ∈ V ↦ (x +c 1c)), G)))
73	71, 72	syl 15	. . . . . 6 ⊢ (¬ I ∈ V → Clos1 ({⟨0c, I⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G)) = Clos1 (∅, PProd ((x ∈ V ↦ (x +c 1c)), G)))
74		eqid 2353	. . . . . . 7 ⊢ Clos1 (∅, PProd ((x ∈ V ↦ (x +c 1c)), G)) = Clos1 (∅, PProd ((x ∈ V ↦ (x +c 1c)), G))
75	61, 74	clos10 5888	. . . . . 6 ⊢ Clos1 (∅, PProd ((x ∈ V ↦ (x +c 1c)), G)) = ∅
76	73, 75	syl6eq 2401	. . . . 5 ⊢ (¬ I ∈ V → Clos1 ({⟨0c, I⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G)) = ∅)
77		0ss 3580	. . . . 5 ⊢ ∅ ⊆ ( Nn × (ran G ∪ {I}))
78	76, 77	syl6eqss 3322	. . . 4 ⊢ (¬ I ∈ V → Clos1 ({⟨0c, I⟩}, PProd ((x ∈ V ↦ (x +c 1c)), G)) ⊆ ( Nn × (ran G ∪ {I})))
79	66, 78	syl5eqss 3316	. . 3 ⊢ (¬ I ∈ V → FRec (G, I) ⊆ ( Nn × (ran G ∪ {I})))
80	65, 79	pm2.61i 156	. 2 ⊢ FRec (G, I) ⊆ ( Nn × (ran G ∪ {I}))
81	1, 80	eqsstri 3302	1 ⊢ F ⊆ ( Nn × (ran G ∪ {I}))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ∪ cun 3208   ⊆ wss 3258  ∅c0 3551  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376  ⟨cop 4562   class class class wbr 4640   × cxp 4771  ran crn 4774   ↦ cmpt 5652   PProd cpprod 5738   Clos1 cclos1 5873   FRec cfrec 6310

This proof depends on 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof 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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-pprod 5739  df-fix 5741  df-cup 5743  df-disj 5745  df-addcfn 5747  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-clos1 5874  df-frec 6311

This theorem is used by:  frecxpg  6316