Theorem zfz1iso 10812

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 6756	. . . 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 4005	. . . . . . . . 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 1824	. . . . . 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 4005	. . . . . . . . 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 1824	. . . . . 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 4005	. . . . . . . . 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 1824	. . . . . 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 4005	. . . . . . . . 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 1824	. . . . . 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 5813	. . . . . . . . . 10 ⊢ ∅ Isom < , < (∅, ∅)
22		en0 6790	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
23	22	biimpi 120	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≈ ∅ → 𝑥 = ∅)
24	23	fveq2d 5516	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≈ ∅ → (♯‘𝑥) = (♯‘∅))
25		hash0 10767	. . . . . . . . . . . . . . 15 ⊢ (♯‘∅) = 0
26	24, 25	eqtrdi 2226	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≈ ∅ → (♯‘𝑥) = 0)
27	26	oveq2d 5886	. . . . . . . . . . . . 13 ⊢ (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = (1...0))
28		fz10 10039	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
29	27, 28	eqtrdi 2226	. . . . . . . . . . . 12 ⊢ (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = ∅)
30		isoeq4 5800	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝑥)) = ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
31	29, 30	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
32		isoeq5 5801	. . . . . . . . . . . 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 4128	. . . . . . . . . 10 ⊢ ∅ ∈ V
37		isoeq1 5797	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥)))
38	36, 37	spcev 2832	. . . . . . . . 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 1449	. . . . . 6 ⊢ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
42		sseq1 3178	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ⊆ ℤ ↔ 𝑥 ⊆ ℤ))
43		eleq1 2240	. . . . . . . . . . 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 4004	. . . . . . . . . 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 5512	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (♯‘𝑎) = (♯‘𝑥))
48	47	oveq2d 5886	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (1...(♯‘𝑎)) = (1...(♯‘𝑥)))
49		isoeq4 5800	. . . . . . . . . . . 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 5801	. . . . . . . . . . 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 1825	. . . . . . . . 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 1917	. . . . . . 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 9621	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℚ
58	56, 57	sstrdi 3167	. . . . . . . . . . . 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 2740	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
61		nsuceq0g 4416	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ V → suc 𝑘 ≠ ∅)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑘 ≠ ∅
63	62	neii 2349	. . . . . . . . . . . . . 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 6778	. . . . . . . . . . . . . . . . 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 4027	. . . . . . . . . . . . . . . 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 6790	. . . . . . . . . . . . . . . 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 664	. . . . . . . . . . . . 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 2354	. . . . . . . . . . . 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 10798	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℚ ∧ 𝑥 ∈ Fin ∧ 𝑥 ≠ ∅) → ∃𝑚 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)
74	58, 59, 72, 73	syl3anc 1238	. . . . . . . . . . 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 10811	. . . . . . . . . . 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 2599	. . . . . . . . . 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 1874	. . . . . . . 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 4597	. . . . 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 3178	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ℤ ↔ 𝐴 ⊆ ℤ))
92		eleq1 2240	. . . . . . . 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 4004	. . . . . . 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 5512	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
97	96	oveq2d 5886	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (1...(♯‘𝑥)) = (1...(♯‘𝐴)))
98		isoeq4 5800	. . . . . . . . 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 5801	. . . . . . . 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 1825	. . . . . 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 2824	. . . 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 2599	1 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃wrex 2456  Vcvv 2737   ⊆ wss 3129  ∅c0 3422   class class class wbr 4001  suc csuc 4363  ωcom 4587  ‘cfv 5213   Isom wiso 5214  (class class class)co 5870   ≈ cen 6733  Fincfn 6735  0cc0 7806  1c1 7807   < clt 7986   ≤ cle 7987  ℤcz 9247  ℚcq 9613  ...cfz 10002  ♯chash 10746

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 4116  ax-sep 4119  ax-nul 4127  ax-pow 4172  ax-pr 4207  ax-un 4431  ax-setind 4534  ax-iinf 4585  ax-cnex 7897  ax-resscn 7898  ax-1cn 7899  ax-1re 7900  ax-icn 7901  ax-addcl 7902  ax-addrcl 7903  ax-mulcl 7904  ax-mulrcl 7905  ax-addcom 7906  ax-mulcom 7907  ax-addass 7908  ax-mulass 7909  ax-distr 7910  ax-i2m1 7911  ax-0lt1 7912  ax-1rid 7913  ax-0id 7914  ax-rnegex 7915  ax-precex 7916  ax-cnre 7917  ax-pre-ltirr 7918  ax-pre-ltwlin 7919  ax-pre-lttrn 7920  ax-pre-apti 7921  ax-pre-ltadd 7922  ax-pre-mulgt0 7923  ax-pre-mulext 7924

This theorem depends on definitions:  df-bi 117  df-dc 835  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-rmo 2463  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-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-int 3844  df-iun 3887  df-br 4002  df-opab 4063  df-mpt 4064  df-tr 4100  df-id 4291  df-po 4294  df-iso 4295  df-iord 4364  df-on 4366  df-ilim 4367  df-suc 4369  df-iom 4588  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-res 4636  df-ima 4637  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-f1 5218  df-fo 5219  df-f1o 5220  df-fv 5221  df-isom 5222  df-riota 5826  df-ov 5873  df-oprab 5874  df-mpo 5875  df-1st 6136  df-2nd 6137  df-recs 6301  df-irdg 6366  df-frec 6387  df-1o 6412  df-oadd 6416  df-er 6530  df-en 6736  df-dom 6737  df-fin 6738  df-pnf 7988  df-mnf 7989  df-xr 7990  df-ltxr 7991  df-le 7992  df-sub 8124  df-neg 8125  df-reap 8526  df-ap 8533  df-div 8624  df-inn 8914  df-n0 9171  df-z 9248  df-uz 9523  df-q 9614  df-rp 9648  df-fz 10003  df-ihash 10747

This theorem is referenced by:  summodclem2  11381  zsumdc  11383  prodmodclem2  11576  zproddc  11578