Theorem nacsfix 43069

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 6873	. . . . 5 ⊢ (𝐹‘𝑧) ⊆ ∪ ran 𝐹
2		simplrr 778	. . . . 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 3979	. . . 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 1216	. . . . 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 777	. . . . 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 43068	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑧))
8	4, 5, 6, 7	syl3anc 1374	. . . 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 3953	. . 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 3130	. 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 6677	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → ran 𝐹 ⊆ 𝐶)
12	11	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ⊆ 𝐶)
13		elpw2g 5280	. . . . . . . 8 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
14	13	3ad2ant1 1134	. . . . . . 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 3463	. . . . . 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 6670	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → 𝐹 Fn ℕ0)
19	18	3ad2ant2 1135	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → 𝐹 Fn ℕ0)
20		0nn0 12428	. . . . . . 7 ⊢ 0 ∈ ℕ0
21		fnfvelrn 7034	. . . . . . 7 ⊢ ((𝐹 Fn ℕ0 ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ ran 𝐹)
22	19, 20, 21	sylancl 587	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (𝐹‘0) ∈ ran 𝐹)
23	22	ne0d 4296	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ≠ ∅)
24		nn0re 12422	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℝ)
25	24	ad2antrl 729	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑎 ∈ ℝ)
26		nn0re 12422	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℝ)
27	26	ad2antll 730	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑏 ∈ ℝ)
28		simplrr 778	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℕ0)
29		simpll3 1216	. . . . . . . . . . . 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 777	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑎 ∈ ℕ0)
31		nn0z 12524	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℤ)
32		nn0z 12524	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℤ)
33		eluz 12777	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑏 ∈ (ℤ≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
34	31, 32, 33	syl2an 597	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . 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 43068	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ (ℤ≥‘𝑎)) → (𝐹‘𝑎) ⊆ (𝐹‘𝑏))
38	29, 30, 36, 37	syl3anc 1374	. . . . . . . . . . 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 4140	. . . . . . . . . . 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 3994	. . . . . . . . . 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 6842	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (𝐹‘𝑐) = (𝐹‘𝑏))
44	43	sseq2d 3968	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)))
45	44	rspcev 3578	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
46	28, 42, 45	syl2anc 585	. . . . . . . 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 777	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℕ0)
48		simpll3 1216	. . . . . . . . . . . 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 778	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑏 ∈ ℕ0)
50		eluz 12777	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℤ ∧ 𝑎 ∈ ℤ) → (𝑎 ∈ (ℤ≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
51	32, 31, 50	syl2anr 598	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . 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 43068	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑏 ∈ ℕ0 ∧ 𝑎 ∈ (ℤ≥‘𝑏)) → (𝐹‘𝑏) ⊆ (𝐹‘𝑎))
55	48, 49, 53, 54	syl3anc 1374	. . . . . . . . . . 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 4143	. . . . . . . . . . 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 3994	. . . . . . . . . 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 6842	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → (𝐹‘𝑐) = (𝐹‘𝑎))
61	60	sseq2d 3968	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)))
62	61	rspcev 3578	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
63	47, 59, 62	syl2anc 585	. . . . . . . 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 11251	. . . . . . 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 3181	. . . . . 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 4115	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 ∪ 𝑧) = ((𝐹‘𝑎) ∪ 𝑧))
67	66	sseq1d 3967	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑎) → ((𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
68	67	rexbidv 3162	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑎) → (∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
69	68	ralbidv 3161	. . . . . . . . 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 7042	. . . . . . . 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 4116	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑏) → ((𝐹‘𝑎) ∪ 𝑧) = ((𝐹‘𝑎) ∪ (𝐹‘𝑏)))
72	71	sseq1d 3967	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
73	72	rexbidv 3162	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑏) → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
74	73	ralrn 7042	. . . . . . . . . 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 3962	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑐) → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
76	75	rexrn 7041	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℕ0 → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
77	76	ralbidv 3161	. . . . . . . . . 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 3161	. . . . . . . 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 18472	. . . . 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 1345	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (toInc‘ran 𝐹) ∈ Dirset)
85		isnacs3 43067	. . . . . . 7 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦)))
86	85	simprbi 497	. . . . . 6 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
87	86	3ad2ant1 1134	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
88		fveq2 6842	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → (toInc‘𝑦) = (toInc‘ran 𝐹))
89	88	eleq1d 2822	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → ((toInc‘𝑦) ∈ Dirset ↔ (toInc‘ran 𝐹) ∈ Dirset))
90		unieq 4876	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → ∪ 𝑦 = ∪ ran 𝐹)
91		id 22	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → 𝑦 = ran 𝐹)
92	90, 91	eleq12d 2831	. . . . . . 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 3576	. . . . 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 585	. . . 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 6902	. . . 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 3154	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 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ∪ cuni 4865   class class class wbr 5100  ran crn 5633   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   ≤ cle 11179  ℕ₀cn0 12413  ℤcz 12500  ℤ_≥cuz 12763  Moorecmre 17513  Dirsetcdrs 18228  toInccipo 18462  NoeACScnacs 43059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-tset 17208  df-ple 17209  df-ocomp 17210  df-mre 17517  df-mrc 17518  df-acs 17520  df-proset 18229  df-drs 18230  df-poset 18248  df-ipo 18463  df-nacs 43060

This theorem is referenced by:  hbt  43487