Theorem frecfzennn 10421

Description: The cardinality of a finite set of sequential integers. (See frec2uz0d 10394 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 5880	. . 3 ⊢ (𝑛 = 0 → (1...𝑛) = (1...0))
2		fveq2 5514	. . 3 ⊢ (𝑛 = 0 → (◡𝐺‘𝑛) = (◡𝐺‘0))
3	1, 2	breq12d 4015	. 2 ⊢ (𝑛 = 0 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...0) ≈ (◡𝐺‘0)))
4		oveq2 5880	. . 3 ⊢ (𝑛 = 𝑚 → (1...𝑛) = (1...𝑚))
5		fveq2 5514	. . 3 ⊢ (𝑛 = 𝑚 → (◡𝐺‘𝑛) = (◡𝐺‘𝑚))
6	4, 5	breq12d 4015	. 2 ⊢ (𝑛 = 𝑚 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑚) ≈ (◡𝐺‘𝑚)))
7		oveq2 5880	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (1...𝑛) = (1...(𝑚 + 1)))
8		fveq2 5514	. . 3 ⊢ (𝑛 = (𝑚 + 1) → (◡𝐺‘𝑛) = (◡𝐺‘(𝑚 + 1)))
9	7, 8	breq12d 4015	. 2 ⊢ (𝑛 = (𝑚 + 1) → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
10		oveq2 5880	. . 3 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
11		fveq2 5514	. . 3 ⊢ (𝑛 = 𝑁 → (◡𝐺‘𝑛) = (◡𝐺‘𝑁))
12	10, 11	breq12d 4015	. 2 ⊢ (𝑛 = 𝑁 → ((1...𝑛) ≈ (◡𝐺‘𝑛) ↔ (1...𝑁) ≈ (◡𝐺‘𝑁)))
13		0ex 4129	. . . 4 ⊢ ∅ ∈ V
14	13	enref 6762	. . 3 ⊢ ∅ ≈ ∅
15		fz10 10041	. . 3 ⊢ (1...0) = ∅
16		0zd 9261	. . . . . . 7 ⊢ (⊤ → 0 ∈ ℤ)
17		frecfzennn.1	. . . . . . 7 ⊢ 𝐺 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
18	16, 17	frec2uzf1od 10401	. . . . . 6 ⊢ (⊤ → 𝐺:ω–1-1-onto→(ℤ≥‘0))
19	18	mptru 1362	. . . . 5 ⊢ 𝐺:ω–1-1-onto→(ℤ≥‘0)
20		peano1 4592	. . . . 5 ⊢ ∅ ∈ ω
21	19, 20	pm3.2i 272	. . . 4 ⊢ (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ ∅ ∈ ω)
22	16, 17	frec2uz0d 10394	. . . . 5 ⊢ (⊤ → (𝐺‘∅) = 0)
23	22	mptru 1362	. . . 4 ⊢ (𝐺‘∅) = 0
24		f1ocnvfv 5777	. . . 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 4032	. 2 ⊢ (1...0) ≈ (◡𝐺‘0)
27		simpr 110	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...𝑚) ≈ (◡𝐺‘𝑚))
28		peano2nn0 9212	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝑚 + 1) ∈ ℕ0)
29		zex 9258	. . . . . . . . . . . . . . 15 ⊢ ℤ ∈ V
30	29	mptex 5741	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℤ ↦ (𝑥 + 1)) ∈ V
31		vex 2740	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
32	30, 31	fvex 5534	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
33	32	ax-gen 1449	. . . . . . . . . . . 12 ⊢ ∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V
34		0z 9260	. . . . . . . . . . . 12 ⊢ 0 ∈ ℤ
35		frecfnom 6399	. . . . . . . . . . . 12 ⊢ ((∀𝑧((𝑥 ∈ ℤ ↦ (𝑥 + 1))‘𝑧) ∈ V ∧ 0 ∈ ℤ) → frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
36	33, 34, 35	mp2an 426	. . . . . . . . . . 11 ⊢ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω
37	17	fneq1i 5309	. . . . . . . . . . 11 ⊢ (𝐺 Fn ω ↔ frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0) Fn ω)
38	36, 37	mpbir 146	. . . . . . . . . 10 ⊢ 𝐺 Fn ω
39		omex 4591	. . . . . . . . . 10 ⊢ ω ∈ V
40		fnex 5737	. . . . . . . . . 10 ⊢ ((𝐺 Fn ω ∧ ω ∈ V) → 𝐺 ∈ V)
41	38, 39, 40	mp2an 426	. . . . . . . . 9 ⊢ 𝐺 ∈ V
42	41	cnvex 5166	. . . . . . . 8 ⊢ ◡𝐺 ∈ V
43		vex 2740	. . . . . . . 8 ⊢ 𝑚 ∈ V
44	42, 43	fvex 5534	. . . . . . 7 ⊢ (◡𝐺‘𝑚) ∈ V
45		en2sn 6810	. . . . . . 7 ⊢ (((𝑚 + 1) ∈ ℕ0 ∧ (◡𝐺‘𝑚) ∈ V) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
46	28, 44, 45	sylancl 413	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
47	46	adantr 276	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → {(𝑚 + 1)} ≈ {(◡𝐺‘𝑚)})
48		fzp1disj 10075	. . . . . 6 ⊢ ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅
49	48	a1i 9	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∩ {(𝑚 + 1)}) = ∅)
50		f1ocnvdm 5779	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (◡𝐺‘𝑚) ∈ ω)
51	19, 50	mpan 424	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘0) → (◡𝐺‘𝑚) ∈ ω)
52		nn0uz 9558	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
53	51, 52	eleq2s 2272	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘𝑚) ∈ ω)
54		nnord 4610	. . . . . . . 8 ⊢ ((◡𝐺‘𝑚) ∈ ω → Ord (◡𝐺‘𝑚))
55		ordirr 4540	. . . . . . . 8 ⊢ (Ord (◡𝐺‘𝑚) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
56	53, 54, 55	3syl 17	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
57	56	adantr 276	. . . . . 6 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
58		disjsn 3654	. . . . . 6 ⊢ (((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅ ↔ ¬ (◡𝐺‘𝑚) ∈ (◡𝐺‘𝑚))
59	57, 58	sylibr 134	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((◡𝐺‘𝑚) ∩ {(◡𝐺‘𝑚)}) = ∅)
60		unen 6813	. . . . 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 1239	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → ((1...𝑚) ∪ {(𝑚 + 1)}) ≈ ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
62		1z 9275	. . . . . 6 ⊢ 1 ∈ ℤ
63		1m1e0 8984	. . . . . . . . . 10 ⊢ (1 − 1) = 0
64	63	fveq2i 5517	. . . . . . . . 9 ⊢ (ℤ≥‘(1 − 1)) = (ℤ≥‘0)
65	52, 64	eqtr4i 2201	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘(1 − 1))
66	65	eleq2i 2244	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘(1 − 1)))
67	66	biimpi 120	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘(1 − 1)))
68		fzsuc2 10074	. . . . . 6 ⊢ ((1 ∈ ℤ ∧ 𝑚 ∈ (ℤ≥‘(1 − 1))) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
69	62, 67, 68	sylancr 414	. . . . 5 ⊢ (𝑚 ∈ ℕ0 → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
70	69	adantr 276	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) = ((1...𝑚) ∪ {(𝑚 + 1)}))
71		peano2 4593	. . . . . . . . 9 ⊢ ((◡𝐺‘𝑚) ∈ ω → suc (◡𝐺‘𝑚) ∈ ω)
72	53, 71	syl 14	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → suc (◡𝐺‘𝑚) ∈ ω)
73	72, 19	jctil 312	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ suc (◡𝐺‘𝑚) ∈ ω))
74		0zd 9261	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑚) ∈ ω → 0 ∈ ℤ)
75		id 19	. . . . . . . . . 10 ⊢ ((◡𝐺‘𝑚) ∈ ω → (◡𝐺‘𝑚) ∈ ω)
76	74, 17, 75	frec2uzsucd 10396	. . . . . . . . 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 2244	. . . . . . . . . . 11 ⊢ (𝑚 ∈ ℕ0 ↔ 𝑚 ∈ (ℤ≥‘0))
79	78	biimpi 120	. . . . . . . . . 10 ⊢ (𝑚 ∈ ℕ0 → 𝑚 ∈ (ℤ≥‘0))
80		f1ocnvfv2 5776	. . . . . . . . . 10 ⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘0) ∧ 𝑚 ∈ (ℤ≥‘0)) → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
81	19, 79, 80	sylancr 414	. . . . . . . . 9 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘(◡𝐺‘𝑚)) = 𝑚)
82	81	oveq1d 5887	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ0 → ((𝐺‘(◡𝐺‘𝑚)) + 1) = (𝑚 + 1))
83	77, 82	eqtrd 2210	. . . . . . 7 ⊢ (𝑚 ∈ ℕ0 → (𝐺‘suc (◡𝐺‘𝑚)) = (𝑚 + 1))
84		f1ocnvfv 5777	. . . . . . 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 62	. . . . . 6 ⊢ (𝑚 ∈ ℕ0 → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
86	85	adantr 276	. . . . 5 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = suc (◡𝐺‘𝑚))
87		df-suc 4370	. . . . 5 ⊢ suc (◡𝐺‘𝑚) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)})
88	86, 87	eqtrdi 2226	. . . 4 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (◡𝐺‘(𝑚 + 1)) = ((◡𝐺‘𝑚) ∪ {(◡𝐺‘𝑚)}))
89	61, 70, 88	3brtr4d 4034	. . 3 ⊢ ((𝑚 ∈ ℕ0 ∧ (1...𝑚) ≈ (◡𝐺‘𝑚)) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1)))
90	89	ex 115	. 2 ⊢ (𝑚 ∈ ℕ0 → ((1...𝑚) ≈ (◡𝐺‘𝑚) → (1...(𝑚 + 1)) ≈ (◡𝐺‘(𝑚 + 1))))
91	3, 6, 9, 12, 26, 90	nn0ind 9363	1 ⊢ (𝑁 ∈ ℕ0 → (1...𝑁) ≈ (◡𝐺‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104  ∀wal 1351   = wceq 1353  ⊤wtru 1354   ∈ wcel 2148  Vcvv 2737   ∪ cun 3127   ∩ cin 3128  ∅c0 3422  {csn 3592   class class class wbr 4002   ↦ cmpt 4063  Ord word 4361  suc csuc 4364  ωcom 4588  ^◡ccnv 4624   Fn wfn 5210  –1-1-onto→wf1o 5214  ‘cfv 5215  (class class class)co 5872  freccfrec 6388   ≈ cen 6735  0cc0 7808  1c1 7809   + caddc 7811   − cmin 8124  ℕ₀cn0 9172  ℤcz 9249  ℤ_≥cuz 9524  ...cfz 10004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-distr 7912  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-cnre 7919  ax-pre-ltirr 7920  ax-pre-ltwlin 7921  ax-pre-lttrn 7922  ax-pre-apti 7923  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-recs 6303  df-frec 6389  df-1o 6414  df-er 6532  df-en 6738  df-pnf 7990  df-mnf 7991  df-xr 7992  df-ltxr 7993  df-le 7994  df-sub 8126  df-neg 8127  df-inn 8916  df-n0 9173  df-z 9250  df-uz 9525  df-fz 10005

This theorem is referenced by:  frecfzen2  10422  hashfz1  10756