Theorem frecfzennn 9419

Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 9401 for a description of the hypothesis.) (Contributed by Jim Kingdon, 18-May-2020.)

Hypothesis

Ref	Expression
frecfzennn.1	⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)

Assertion

Ref	Expression
frecfzennn	⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))

Proof of Theorem frecfzennn

Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 5540	. . 3 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
2		fveq2 5198	. . 3 ⊢ (𝑛 = 0 → (◡𝐺‘𝑛) = (◡𝐺‘0))
3	1, 2	breq12d 3798	. 2 ⊢ (𝑛 = 0 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...0) ≈ (◡𝐺‘0)))
4		oveq2 5540	. . 3 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5		fveq2 5198	. . 3 ⊢ (𝑛 = 𝑚 → (◡𝐺‘𝑛) = (◡𝐺‘𝑚))
6	4, 5	breq12d 3798	. 2 ⊢ (𝑛 = 𝑚 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑚) ≈ (◡𝐺‘𝑚)))
7		oveq2 5540	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8		fveq2 5198	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (◡𝐺‘𝑛) = (◡𝐺‘(𝑚 + 1)))
9	7, 8	breq12d 3798	. 2 ⊢ (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
10		oveq2 5540	. . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11		fveq2 5198	. . 3 ⊢ (𝑛 = 𝑁 → (◡𝐺‘𝑛) = (◡𝐺‘𝑁))
12	10, 11	breq12d 3798	. 2 ⊢ (𝑛 = 𝑁 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑁) ≈ (◡𝐺‘𝑁)))
13		0ex 3905	. . . 4 ⊢ ∅ ∈ V
14	13	enref 6268	. . 3 ⊢ ∅ ≈ ∅
15		fz10 9065	. . 3 ⊢ (1...0) = ∅
16		0zd 8363	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℤ)
17		frecfzennn.1	. . . . . . 7 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
18	16, 17	frec2uzf1od 9408	. . . . . 6 ⊢ (⊤ → 𝐺:ω–1-1-onto→(ℤ≥‘0))
19	18	trud 1293	. . . . 5 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘0)
20		peano1 4335	. . . . 5 ⊢ ∅ ∈ ω
21	19, 20	pm3.2i 266	. . . 4 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ ∅ ∈ ω)
22	16, 17	frec2uz0d 9401	. . . . 5 ⊢ (⊤ → (𝐺‘∅) = 0)
23	22	trud 1293	. . . 4 ⊢ (𝐺‘∅) = 0
24		f1ocnvfv 5439	. . . 4 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ ∅ ∈ ω) → ((𝐺‘∅) = 0 → (◡𝐺‘0) = ∅))
25	21, 23, 24	mp2 16	. . 3 ⊢ (◡𝐺‘0) = ∅
26	14, 15, 25	3brtr4i 3813	. 2 ⊢ (1...0) ≈ (◡𝐺‘0)
27		simpr 108	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...𝑚) ≈ (◡𝐺‘𝑚))
28		peano2nn0 8328	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
29		zex 8360	. . . . . . . . . . . . . . 15 ⊢ ℤ ∈ V
30	29	mptex 5408	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
31		vex 2604	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
32	30, 31	fvex 5215	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
33	32	ax-gen 1378	. . . . . . . . . . . 12 ⊢ ∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
34		0z 8362	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
35		frecfnom 6009	. . . . . . . . . . . 12 ⊢ ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
36	33, 34, 35	mp2an 416	. . . . . . . . . . 11 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
37	17	fneq1i 5013	. . . . . . . . . . 11 ⊢ (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
38	36, 37	mpbir 144	. . . . . . . . . 10 ⊢ 𝐺 Fn ω
39		omex 4334	. . . . . . . . . 10 ⊢ ω ∈ V
40		fnex 5404	. . . . . . . . . 10 ⊢ ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
41	38, 39, 40	mp2an 416	. . . . . . . . 9 ⊢ 𝐺 ∈ V
42	41	cnvex 4876	. . . . . . . 8 ⊢ ◡𝐺 ∈ V
43		vex 2604	. . . . . . . 8 ⊢ 𝑚 ∈ V
44	42, 43	fvex 5215	. . . . . . 7 ⊢ (◡𝐺‘𝑚) ∈ V
45		en2sn 6313	. . . . . . 7 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ (◡𝐺‘𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
46	28, 44, 45	sylancl 404	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
47	46	adantr 270	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
48		fzp1disj 9097	. . . . . 6 ⊢ ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
49	48	a1i 9	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
50		f1ocnvdm 5441	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (◡𝐺‘𝑚) ∈ ω)
51	19, 50	mpan 414	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘0) → (◡𝐺‘𝑚) ∈ ω)
52		nn0uz 8653	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleq2s 2173	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘𝑚) ∈ ω)
54		nnord 4352	. . . . . . . 8 ⊢ ((◡𝐺‘𝑚) ∈ ω → Ord (◡𝐺‘𝑚))
55		ordirr 4285	. . . . . . . 8 ⊢ (Ord (◡𝐺‘𝑚) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
56	53, 54, 55	3syl 17	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
57	56	adantr 270	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
58		disjsn 3454	. . . . . 6 ⊢ (((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅ ↔ ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
59	57, 58	sylibr 132	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅)
60		unen 6316	. . . . 5 ⊢ ((((1...𝑚) ≈ (◡𝐺‘𝑚) ∧ {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)}) ∧ (((1...𝑚) ∩ {(𝑚 + 1)}) = ∅ ∧ ((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
61	27, 47, 49, 59, 60	syl22anc 1170	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
62		1z 8377	. . . . . 6 ⊢ 1 ∈ ℤ
63		1m1e0 8108	. . . . . . . . . 10 ⊢ (1 − 1) = 0
64	63	fveq2i 5201	. . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
65	52, 64	eqtr4i 2104	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
66	65	eleq2i 2145	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘(1 − 1)))
67	66	biimpi 118	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘(1 − 1)))
68		fzsuc2 9096	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
69	62, 67, 68	sylancr 405	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
70	69	adantr 270	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
71		peano2 4336	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑚) ∈ ω → suc (◡𝐺‘𝑚) ∈ ω)
72	53, 71	syl 14	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → suc (◡𝐺‘𝑚) ∈ ω)
73	72, 19	jctil 305	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ suc (◡𝐺‘𝑚) ∈ ω))
74		0zd 8363	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑚) ∈ ω → 0 ∈ ℤ)
75		id 19	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑚) ∈ ω → (◡𝐺‘𝑚) ∈ ω)
76	74, 17, 75	frec2uzsucd 9403	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑚) ∈ ω → (𝐺‘suc (◡𝐺‘𝑚)) = ((𝐺‘(◡𝐺‘𝑚)) + 1))
77	53, 76	syl 14	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘suc (◡𝐺‘𝑚)) = ((𝐺‘(◡𝐺‘𝑚)) + 1))
78	52	eleq2i 2145	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘0))
79	78	biimpi 118	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘0))
80		f1ocnvfv2 5438	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
81	19, 79, 80	sylancr 405	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
82	81	oveq1d 5547	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → ((𝐺‘(◡𝐺‘𝑚)) + 1) = (𝑚 + 1))
83	77, 82	eqtrd 2113	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘suc (◡𝐺‘𝑚)) = (𝑚 + 1))
84		f1ocnvfv 5439	. . . . . . 7 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ suc (◡𝐺‘𝑚) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝑚)) = (𝑚 + 1) → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚)))
85	73, 83, 84	sylc 61	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
86	85	adantr 270	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
87		df-suc 4126	. . . . 5 ⊢ suc (◡𝐺‘𝑚) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)})
88	86, 87	syl6eq 2129	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
89	61, 70, 88	3brtr4d 3815	. . 3 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1)))
90	89	ex 113	. 2 ⊢ (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (◡𝐺‘𝑚) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
91	3, 6, 9, 12, 26, 90	nn0ind 8461	1 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102  ∀wal 1282   = wceq 1284  ⊤wtru 1285   ∈ wcel 1433  Vcvv 2601   ∪ cun 2971   ∩ cin 2972  ∅c0 3251  {csn 3398   class class class wbr 3785   ↦ cmpt 3839  Ord word 4117  suc csuc 4120  ωcom 4331  ^◡ccnv 4362   Fn wfn 4917  –1-1-onto→wf1o 4921  ‘cfv 4922  (class class class)co 5532  freccfrec 6000   ≈ cen 6242  0cc0 6981  1c1 6982   + caddc 6984   − cmin 7279  ℕ₀cn0 8288  ℤcz 8351  ℤ_≥cuz 8619  ...cfz 9029

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-0id 7084  ax-rnegex 7085  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-recs 5943  df-frec 6001  df-1o 6024  df-er 6129  df-en 6245  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-fz 9030

This theorem is referenced by:  frecfzen2  9420