Theorem nacsfix 42702

Description: An increasing sequence of closed sets in a Noetherian-type closure system eventually fixates. (Contributed by Stefan O'Rear, 4-Apr-2015.)

Assertion

Ref	Expression
nacsfix	⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 ∀𝑧 ∈ (ℤ≥‘𝑦)(𝐹‘𝑧) = (𝐹‘𝑦))

Distinct variable groups:   𝑧,𝐶,𝑦   𝑦,𝐹,𝑧   𝑧,𝑋,𝑦   𝑥,𝑦,𝑧,𝐹

Allowed substitution hints:   𝐶(𝑥)   𝑋(𝑥)

Proof of Theorem nacsfix

Dummy variables 𝑎 𝑏 𝑐 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvssunirn 6914	. . . . 5 ⊢ (𝐹‘𝑧) ⊆ ∪ ran 𝐹
2		simplrr 777	. . . . 5 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑦) = ∪ ran 𝐹)
3	1, 2	sseqtrrid 4007	. . . 4 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑧) ⊆ (𝐹‘𝑦))
4		simpll3 1215	. . . . 5 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)))
5		simplrl 776	. . . . 5 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → 𝑦 ∈ ℕ0)
6		simpr 484	. . . . 5 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → 𝑧 ∈ (ℤ≥‘𝑦))
7		incssnn0 42701	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑧))
8	4, 5, 6, 7	syl3anc 1373	. . . 4 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑧))
9	3, 8	eqssd 3981	. . 3 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑧) = (𝐹‘𝑦))
10	9	ralrimiva 3133	. 2 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹‘𝑦) = ∪ ran 𝐹)) → ∀𝑧 ∈ (ℤ≥‘𝑦)(𝐹‘𝑧) = (𝐹‘𝑦))
11		frn 6718	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → ran 𝐹 ⊆ 𝐶)
12	11	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ⊆ 𝐶)
13		elpw2g 5308	. . . . . . . 8 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
14	13	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
15	12, 14	mpbird 257	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ∈ 𝒫 𝐶)
16		elex 3485	. . . . . 6 ⊢ (ran 𝐹 ∈ 𝒫 𝐶 → ran 𝐹 ∈ V)
17	15, 16	syl 17	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ∈ V)
18		ffn 6711	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → 𝐹 Fn ℕ0)
19	18	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → 𝐹 Fn ℕ0)
20		0nn0 12521	. . . . . . 7 ⊢ 0 ∈ ℕ0
21		fnfvelrn 7075	. . . . . . 7 ⊢ ((𝐹 Fn ℕ0 ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ ran 𝐹)
22	19, 20, 21	sylancl 586	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (𝐹‘0) ∈ ran 𝐹)
23	22	ne0d 4322	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ≠ ∅)
24		nn0re 12515	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℝ)
25	24	ad2antrl 728	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑎 ∈ ℝ)
26		nn0re 12515	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℝ)
27	26	ad2antll 729	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑏 ∈ ℝ)
28		simplrr 777	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℕ0)
29		simpll3 1215	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)))
30		simplrl 776	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑎 ∈ ℕ0)
31		nn0z 12618	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℤ)
32		nn0z 12618	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℤ)
33		eluz 12871	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑏 ∈ (ℤ≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
34	31, 32, 33	syl2an 596	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑏 ∈ (ℤ≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
35	34	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ (ℤ≥‘𝑎))
36	35	adantll 714	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ (ℤ≥‘𝑎))
37		incssnn0 42701	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ (ℤ≥‘𝑎)) → (𝐹‘𝑎) ⊆ (𝐹‘𝑏))
38	29, 30, 36, 37	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → (𝐹‘𝑎) ⊆ (𝐹‘𝑏))
39		ssequn1 4166	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑎) ⊆ (𝐹‘𝑏) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑏))
40	38, 39	sylib 218	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑏))
41		eqimss 4022	. . . . . . . . . 10 ⊢ (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑏) → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏))
42	40, 41	syl 17	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏))
43		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (𝐹‘𝑐) = (𝐹‘𝑏))
44	43	sseq2d 3996	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)))
45	44	rspcev 3606	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
46	28, 42, 45	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
47		simplrl 776	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℕ0)
48		simpll3 1215	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)))
49		simplrr 777	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑏 ∈ ℕ0)
50		eluz 12871	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℤ ∧ 𝑎 ∈ ℤ) → (𝑎 ∈ (ℤ≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
51	32, 31, 50	syl2anr 597	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 ∈ (ℤ≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
52	51	biimpar 477	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ (ℤ≥‘𝑏))
53	52	adantll 714	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ (ℤ≥‘𝑏))
54		incssnn0 42701	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑏 ∈ ℕ0 ∧ 𝑎 ∈ (ℤ≥‘𝑏)) → (𝐹‘𝑏) ⊆ (𝐹‘𝑎))
55	48, 49, 53, 54	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → (𝐹‘𝑏) ⊆ (𝐹‘𝑎))
56		ssequn2 4169	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑏) ⊆ (𝐹‘𝑎) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑎))
57	55, 56	sylib 218	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑎))
58		eqimss 4022	. . . . . . . . . 10 ⊢ (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑎) → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎))
59	57, 58	syl 17	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎))
60		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → (𝐹‘𝑐) = (𝐹‘𝑎))
61	60	sseq2d 3996	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)))
62	61	rspcev 3606	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
63	47, 59, 62	syl2anc 584	. . . . . . . 8 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
64	25, 27, 46, 63	lecasei 11346	. . . . . . 7 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
65	64	ralrimivva 3188	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑎 ∈ ℕ0 ∀𝑏 ∈ ℕ0 ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
66		uneq1 4141	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 ∪ 𝑧) = ((𝐹‘𝑎) ∪ 𝑧))
67	66	sseq1d 3995	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑎) → ((𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
68	67	rexbidv 3165	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑎) → (∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
69	68	ralbidv 3164	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑎) → (∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
70	69	ralrn 7083	. . . . . . . 8 ⊢ (𝐹 Fn ℕ0 → (∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0 ∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
71		uneq2 4142	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑏) → ((𝐹‘𝑎) ∪ 𝑧) = ((𝐹‘𝑎) ∪ (𝐹‘𝑏)))
72	71	sseq1d 3995	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
73	72	rexbidv 3165	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑏) → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
74	73	ralrn 7083	. . . . . . . . . 10 ⊢ (𝐹 Fn ℕ0 → (∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑏 ∈ ℕ0 ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
75		sseq2 3990	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑐) → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
76	75	rexrn 7082	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℕ0 → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
77	76	ralbidv 3164	. . . . . . . . . 10 ⊢ (𝐹 Fn ℕ0 → (∀𝑏 ∈ ℕ0 ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ∀𝑏 ∈ ℕ0 ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
78	74, 77	bitrd 279	. . . . . . . . 9 ⊢ (𝐹 Fn ℕ0 → (∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑏 ∈ ℕ0 ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
79	78	ralbidv 3164	. . . . . . . 8 ⊢ (𝐹 Fn ℕ0 → (∀𝑎 ∈ ℕ0 ∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0 ∀𝑏 ∈ ℕ0 ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
80	70, 79	bitrd 279	. . . . . . 7 ⊢ (𝐹 Fn ℕ0 → (∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0 ∀𝑏 ∈ ℕ0 ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
81	19, 80	syl 17	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0 ∀𝑏 ∈ ℕ0 ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
82	65, 81	mpbird 257	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤)
83		isipodrs 18552	. . . . 5 ⊢ ((toInc‘ran 𝐹) ∈ Dirset ↔ (ran 𝐹 ∈ V ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤))
84	17, 23, 82, 83	syl3anbrc 1344	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (toInc‘ran 𝐹) ∈ Dirset)
85		isnacs3 42700	. . . . . . 7 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦)))
86	85	simprbi 496	. . . . . 6 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
87	86	3ad2ant1 1133	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
88		fveq2 6881	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → (toInc‘𝑦) = (toInc‘ran 𝐹))
89	88	eleq1d 2820	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → ((toInc‘𝑦) ∈ Dirset ↔ (toInc‘ran 𝐹) ∈ Dirset))
90		unieq 4899	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → ∪ 𝑦 = ∪ ran 𝐹)
91		id 22	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → 𝑦 = ran 𝐹)
92	90, 91	eleq12d 2829	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ ran 𝐹 ∈ ran 𝐹))
93	89, 92	imbi12d 344	. . . . . 6 ⊢ (𝑦 = ran 𝐹 → (((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦) ↔ ((toInc‘ran 𝐹) ∈ Dirset → ∪ ran 𝐹 ∈ ran 𝐹)))
94	93	rspcva 3604	. . . . 5 ⊢ ((ran 𝐹 ∈ 𝒫 𝐶 ∧ ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦)) → ((toInc‘ran 𝐹) ∈ Dirset → ∪ ran 𝐹 ∈ ran 𝐹))
95	15, 87, 94	syl2anc 584	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ((toInc‘ran 𝐹) ∈ Dirset → ∪ ran 𝐹 ∈ ran 𝐹))
96	84, 95	mpd 15	. . 3 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∪ ran 𝐹 ∈ ran 𝐹)
97		fvelrnb 6944	. . . 4 ⊢ (𝐹 Fn ℕ0 → (∪ ran 𝐹 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ℕ0 (𝐹‘𝑦) = ∪ ran 𝐹))
98	19, 97	syl 17	. . 3 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (∪ ran 𝐹 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ℕ0 (𝐹‘𝑦) = ∪ ran 𝐹))
99	96, 98	mpbid 232	. 2 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 (𝐹‘𝑦) = ∪ ran 𝐹)
100	10, 99	reximddv 3157	1 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 ∀𝑧 ∈ (ℤ≥‘𝑦)(𝐹‘𝑧) = (𝐹‘𝑦))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  ∪ cuni 4888   class class class wbr 5124  ran crn 5660   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  ℝcr 11133  0cc0 11134  1c1 11135   + caddc 11137   ≤ cle 11275  ℕ₀cn0 12506  ℤcz 12593  ℤ_≥cuz 12857  Moorecmre 17599  Dirsetcdrs 18310  toInccipo 18542  NoeACScnacs 42692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12858  df-fz 13530  df-struct 17171  df-slot 17206  df-ndx 17218  df-base 17234  df-tset 17295  df-ple 17296  df-ocomp 17297  df-mre 17603  df-mrc 17604  df-acs 17606  df-proset 18311  df-drs 18312  df-poset 18330  df-ipo 18543  df-nacs 42693

This theorem is referenced by:  hbt  43121