Theorem zfz1iso 11293

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 7047	. . . 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 544	. . . 4 ⊢ ((𝑛 ∈ ω ∧ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛)) → 𝐴 ∈ Fin)
5		breq2 4134	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ ∅))
6	5	anbi2d 468	. . . . . . . 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 1877	. . . . . 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 4134	. . . . . . . . 9 ⊢ (𝑤 = 𝑘 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑘))
10	9	anbi2d 468	. . . . . . . 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 1877	. . . . . 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 4134	. . . . . . . . 9 ⊢ (𝑤 = suc 𝑘 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ suc 𝑘))
14	13	anbi2d 468	. . . . . . . 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 1877	. . . . . 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 4134	. . . . . . . . 9 ⊢ (𝑤 = 𝑛 → (𝑥 ≈ 𝑤 ↔ 𝑥 ≈ 𝑛))
18	17	anbi2d 468	. . . . . . . 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 1877	. . . . . 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 6023	. . . . . . . . . 10 ⊢ ∅ Isom < , < (∅, ∅)
22		en0 7082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
23	22	biimpi 120	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ≈ ∅ → 𝑥 = ∅)
24	23	fveq2d 5699	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ≈ ∅ → (♯‘𝑥) = (♯‘∅))
25		hash0 11235	. . . . . . . . . . . . . . 15 ⊢ (♯‘∅) = 0
26	24, 25	eqtrdi 2287	. . . . . . . . . . . . . 14 ⊢ (𝑥 ≈ ∅ → (♯‘𝑥) = 0)
27	26	oveq2d 6101	. . . . . . . . . . . . 13 ⊢ (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = (1...0))
28		fz10 10450	. . . . . . . . . . . . 13 ⊢ (1...0) = ∅
29	27, 28	eqtrdi 2287	. . . . . . . . . . . 12 ⊢ (𝑥 ≈ ∅ → (1...(♯‘𝑥)) = ∅)
30		isoeq4 6010	. . . . . . . . . . . 12 ⊢ ((1...(♯‘𝑥)) = ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
31	29, 30	syl 14	. . . . . . . . . . 11 ⊢ (𝑥 ≈ ∅ → (∅ Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < (∅, 𝑥)))
32		isoeq5 6011	. . . . . . . . . . . 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 4260	. . . . . . . . . 10 ⊢ ∅ ∈ V
37		isoeq1 6007	. . . . . . . . . 10 ⊢ (𝑓 = ∅ → (𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥) ↔ ∅ Isom < , < ((1...(♯‘𝑥)), 𝑥)))
38	36, 37	spcev 2920	. . . . . . . . 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 1502	. . . . . 6 ⊢ ∀𝑥(((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ ∅) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝑥)), 𝑥))
42		sseq1 3271	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ⊆ ℤ ↔ 𝑥 ⊆ ℤ))
43		eleq1 2301	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ Fin ↔ 𝑥 ∈ Fin))
44	42, 43	anbi12d 477	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ↔ (𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin)))
45		breq1 4133	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → (𝑎 ≈ 𝑘 ↔ 𝑥 ≈ 𝑘))
46	44, 45	anbi12d 477	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (((𝑎 ⊆ ℤ ∧ 𝑎 ∈ Fin) ∧ 𝑎 ≈ 𝑘) ↔ ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑘)))
47		fveq2 5695	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑥 → (♯‘𝑎) = (♯‘𝑥))
48	47	oveq2d 6101	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (1...(♯‘𝑎)) = (1...(♯‘𝑥)))
49		isoeq4 6010	. . . . . . . . . . . 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 6011	. . . . . . . . . . 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 1878	. . . . . . . . 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 1973	. . . . . . 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 543	. . . . . . . . . . . . 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 10027	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℚ
58	56, 57	sstrdi 3260	. . . . . . . . . . . 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 544	. . . . . . . . . . . 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 2824	. . . . . . . . . . . . . . . 16 ⊢ 𝑘 ∈ V
61		nsuceq0g 4563	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ V → suc 𝑘 ≠ ∅)
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ suc 𝑘 ≠ ∅
63	62	neii 2422	. . . . . . . . . . . . . 14 ⊢ ¬ suc 𝑘 = ∅
64		simplrr 542	. . . . . . . . . . . . . . . . . 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 7070	. . . . . . . . . . . . . . . . 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 4156	. . . . . . . . . . . . . . . 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 7082	. . . . . . . . . . . . . . . 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 674	. . . . . . . . . . . . 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 2427	. . . . . . . . . . . 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 11270	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ ℚ ∧ 𝑥 ∈ Fin ∧ 𝑥 ≠ ∅) → ∃𝑚 ∈ 𝑥 ∀𝑧 ∈ 𝑥 𝑧 ≤ 𝑚)
74	58, 59, 72, 73	syl3anc 1278	. . . . . . . . . . 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 539	. . . . . . . . . . . 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 540	. . . . . . . . . . . 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 542	. . . . . . . . . . . 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 535	. . . . . . . . . . . 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 537	. . . . . . . . . . . 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 11292	. . . . . . . . . . 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 2673	. . . . . . . . . 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 1927	. . . . . . . 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 4747	. . . . 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 3271	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ℤ ↔ 𝐴 ⊆ ℤ))
92		eleq1 2301	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fin ↔ 𝐴 ∈ Fin))
93	91, 92	anbi12d 477	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ↔ (𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin)))
94		breq1 4133	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑛 ↔ 𝐴 ≈ 𝑛))
95	93, 94	anbi12d 477	. . . . . 6 ⊢ (𝑥 = 𝐴 → (((𝑥 ⊆ ℤ ∧ 𝑥 ∈ Fin) ∧ 𝑥 ≈ 𝑛) ↔ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ 𝐴 ≈ 𝑛)))
96		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (♯‘𝑥) = (♯‘𝐴))
97	96	oveq2d 6101	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (1...(♯‘𝑥)) = (1...(♯‘𝐴)))
98		isoeq4 6010	. . . . . . . . 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 6011	. . . . . . . 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 1878	. . . . . 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 2912	. . . 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 571	. 2 ⊢ (((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))
107	3, 106	rexlimddv 2673	1 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , < ((1...(♯‘𝐴)), 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ⊆ wss 3220  ∅c0 3520   class class class wbr 4130  suc csuc 4510  ωcom 4737  ‘cfv 5377   Isom wiso 5378  (class class class)co 6085   ≈ cen 7020  Fincfn 7022  0cc0 8179  1c1 8180   < clt 8360   ≤ cle 8361  ℤcz 9644  ℚcq 10019  ...cfz 10411  ♯chash 11214

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-ihash 11215

This theorem is used by:  summodclem2  12149  zsumdc  12151  prodmodclem2  12344  zproddc  12346