Theorem zfz1iso 10950

Description: A finite set of integers has an order isomorphism to a one-based finite sequence. (Contributed by Jim Kingdon, 3-Sep-2022.)

Assertion

Ref	Expression
zfz1iso	⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))

Distinct variable group:   𝐴,𝑓

Proof of Theorem zfz1iso

Dummy variables 𝑛 𝑥 𝑎 𝑘 𝑚 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 6829	. . . 4 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
2	1	biimpi 120	. . 3 ⊢ (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
3	2	adantl 277	. 2 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
4		simprlr 538	. . . 4 ⊢ ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛)) → 𝐴 ∈ Fin)
5		breq2 4038	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
6	5	anbi2d 464	. . . . . . . 8 ⊢ (𝑤 = ∅ → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅)))
7	6	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = ∅ → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
8	7	albidv 1838	. . . . . 6 ⊢ (𝑤 = ∅ → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
9		breq2 4038	. . . . . . . . 9 ⊢ (𝑤 = 𝑘 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑘))
10	9	anbi2d 464	. . . . . . . 8 ⊢ (𝑤 = 𝑘 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘)))
11	10	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = 𝑘 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
12	11	albidv 1838	. . . . . 6 ⊢ (𝑤 = 𝑘 → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
13		breq2 4038	. . . . . . . . 9 ⊢ (𝑤 = suc 𝑘 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑘))
14	13	anbi2d 464	. . . . . . . 8 ⊢ (𝑤 = suc 𝑘 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)))
15	14	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = suc 𝑘 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
16	15	albidv 1838	. . . . . 6 ⊢ (𝑤 = suc 𝑘 → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
17		breq2 4038	. . . . . . . . 9 ⊢ (𝑤 = 𝑛 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑛))
18	17	anbi2d 464	. . . . . . . 8 ⊢ (𝑤 = 𝑛 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛)))
19	18	imbi1d 231	. . . . . . 7 ⊢ (𝑤 = 𝑛 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
20	19	albidv 1838	. . . . . 6 ⊢ (𝑤 = 𝑛 → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑤) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
21		iso0 5867	. . . . . . . . . 10 ⊢ ∅ Isom < , < (∅, ∅)
22		en0 6863	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
23	22	biimpi 120	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≈ ∅ → 𝑥 = ∅)
24	23	fveq2d 5565	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≈ ∅ → (♯‘𝑥) = (♯‘∅))
25		hash0 10905	. . . . . . . . . . . . . . 15 ⊢ (♯‘∅) = 0
26	24, 25	eqtrdi 2245	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≈ ∅ → (♯‘𝑥) = 0)
27	26	oveq2d 5941	. . . . . . . . . . . . 13 ⊢ (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = (1...0))
28		fz10 10138	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
29	27, 28	eqtrdi 2245	. . . . . . . . . . . 12 ⊢ (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = ∅)
30		isoeq4 5854	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝑥)) = ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
31	29, 30	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
32		isoeq5 5855	. . . . . . . . . . . 12 ⊢ (𝑥 = ∅ → (∅ Isom < , < (∅, 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
33	23, 32	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 ≈ ∅ → (∅ Isom < , < (∅, 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
34	31, 33	bitrd 188	. . . . . . . . . 10 ⊢ (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, ∅)))
35	21, 34	mpbiri 168	. . . . . . . . 9 ⊢ (𝑥 ≈ ∅ → ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥))
36		0ex 4161	. . . . . . . . . 10 ⊢ ∅ ∈ V
37		isoeq1 5851	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥)))
38	36, 37	spcev 2859	. . . . . . . . 9 ⊢ (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
39	35, 38	syl 14	. . . . . . . 8 ⊢ (𝑥 ≈ ∅ → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
40	39	adantl 277	. . . . . . 7 ⊢ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
41	40	ax-gen 1463	. . . . . 6 ⊢ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
42		sseq1 3207	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ⊆ ℤ ↔ 𝑥 ⊆ ℤ))
43		eleq1 2259	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ Fin ↔ 𝑥 ∈ Fin))
44	42, 43	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ↔ (𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin)))
45		breq1 4037	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 ≈ 𝑘 ↔ 𝑥 ≈ 𝑘))
46	44, 45	anbi12d 473	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘)))
47		fveq2 5561	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (♯‘𝑎) = (♯‘𝑥))
48	47	oveq2d 5941	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (1...(♯‘𝑎)) = (1...(♯‘𝑥)))
49		isoeq4 5854	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝑎)) = (1...(♯‘𝑥)) → (𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑎)))
50	48, 49	syl 14	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑎)))
51		isoeq5 5855	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
52	50, 51	bitrd 188	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
53	52	exbidv 1839	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
54	46, 53	imbi12d 234	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → ((((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)) ↔ (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
55	54	cbvalv 1932	. . . . . . 7 ⊢ (∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)) ↔ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
56		simprll 537	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ⊆ ℤ)
57		zssq 9718	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℚ
58	56, 57	sstrdi 3196	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ⊆ ℚ)
59		simprlr 538	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ∈ Fin)
60		vex 2766	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
61		nsuceq0g 4454	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ V → suc 𝑘 ≠ ∅)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑘 ≠ ∅
63	62	neii 2369	. . . . . . . . . . . . . 14 ⊢ ¬ suc 𝑘 = ∅
64		simplrr 536	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → 𝑥 ≈ suc 𝑘)
65	64	ensymd 6851	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → suc 𝑘 ≈ 𝑥)
66		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → 𝑥 = ∅)
67	65, 66	breqtrd 4060	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → suc 𝑘 ≈ ∅)
68		en0 6863	. . . . . . . . . . . . . . . 16 ⊢ (suc 𝑘 ≈ ∅ ↔ suc 𝑘 = ∅)
69	67, 68	sylib 122	. . . . . . . . . . . . . . 15 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ 𝑥 = ∅) → suc 𝑘 = ∅)
70	69	ex 115	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → (𝑥 = ∅ → suc 𝑘 = ∅))
71	63, 70	mtoi 665	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → ¬ 𝑥 = ∅)
72	71	neqned 2374	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → 𝑥 ≠ ∅)
73		fimaxq 10936	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℚ ∧ 𝑥 ∈ Fin ∧ 𝑥 ≠ ∅) → ∃𝑚 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)
74	58, 59, 72, 73	syl3anc 1249	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → ∃𝑚 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)
75		simplll 533	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → 𝑘 ∈ ω)
76		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)))
77	56	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → 𝑥 ⊆ ℤ)
78	59	adantr 276	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → 𝑥 ∈ Fin)
79		simplrr 536	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → 𝑥 ≈ suc 𝑘)
80		simprl 529	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → 𝑚 ∈ 𝑥)
81		simprr 531	. . . . . . . . . . . 12 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)
82	75, 76, 77, 78, 79, 80, 81	zfz1isolem1 10949	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) ∧ (𝑚 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
83	74, 82	rexlimddv 2619	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) ∧ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
84	83	ex 115	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
85	84	alrimiv 1888	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ ∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎))) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
86	85	ex 115	. . . . . . 7 ⊢ (𝑘 ∈ ω → (∀𝑎(((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑎)), 𝑎)) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
87	55, 86	biimtrrid 153	. . . . . 6 ⊢ (𝑘 ∈ ω → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ suc 𝑘) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))))
88	8, 12, 16, 20, 41, 87	finds 4637	. . . . 5 ⊢ (𝑛 ∈ ω → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
89	88	adantr 276	. . . 4 ⊢ ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛)) → ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)))
90		simpr 110	. . . 4 ⊢ ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛)) → ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛))
91		sseq1 3207	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ℤ ↔ 𝐴 ⊆ ℤ))
92		eleq1 2259	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fin ↔ 𝐴 ∈ Fin))
93	91, 92	anbi12d 473	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ↔ (𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin)))
94		breq1 4037	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑛 ↔ 𝐴 ≈ 𝑛))
95	93, 94	anbi12d 473	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) ↔ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛)))
96		fveq2 5561	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
97	96	oveq2d 5941	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (1...(♯‘𝑥)) = (1...(♯‘𝐴)))
98		isoeq4 5854	. . . . . . . . 9 ⊢ ((1...(♯‘𝑥)) = (1...(♯‘𝐴)) → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝑥)))
99	97, 98	syl 14	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝑥)))
100		isoeq5 5855	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝐴)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
101	99, 100	bitrd 188	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
102	101	exbidv 1839	. . . . . 6 ⊢ (𝑥 = 𝐴 → (∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴)))
103	95, 102	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝐴 → ((((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) ↔ (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))))
104	103	spcgv 2851	. . . 4 ⊢ (𝐴 ∈ Fin → (∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥)) → (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))))
105	4, 89, 90, 104	syl3c 63	. . 3 ⊢ ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
106	105	an12s 565	. 2 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
107	3, 106	rexlimddv 2619	1 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃wex 1506   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ∃wrex 2476  Vcvv 2763   ⊆ wss 3157  ∅c0 3451   class class class wbr 4034  suc csuc 4401  ωcom 4627  ‘cfv 5259   Isom wiso 5260  (class class class)co 5925   ≈ cen 6806  Fincfn 6808  0cc0 7896  1c1 7897   < clt 8078   ≤ cle 8079  ℤcz 9343  ℚcq 9710  ...cfz 10100  ♯chash 10884

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-en 6809  df-dom 6810  df-fin 6811  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-fz 10101  df-ihash 10885

This theorem is referenced by:  summodclem2  11564  zsumdc  11566  prodmodclem2  11759  zproddc  11761