Theorem nacsfix 41905

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 775	. . . . 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 4027	. . . 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 1211	. . . . 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 774	. . . . 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 41904	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑧))
8	4, 5, 6, 7	syl3anc 1368	. . . 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 3991	. . 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 3138	. 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 6714	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → ran 𝐹 ⊆ 𝐶)
12	11	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ⊆ 𝐶)
13		elpw2g 5334	. . . . . . . 8 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
14	13	3ad2ant1 1130	. . . . . . 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 6707	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → 𝐹 Fn ℕ0)
19	18	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → 𝐹 Fn ℕ0)
20		0nn0 12483	. . . . . . 7 ⊢ 0 ∈ ℕ0
21		fnfvelrn 7072	. . . . . . 7 ⊢ ((𝐹 Fn ℕ0 ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ ran 𝐹)
22	19, 20, 21	sylancl 585	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (𝐹‘0) ∈ ran 𝐹)
23	22	ne0d 4327	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ≠ ∅)
24		nn0re 12477	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℝ)
25	24	ad2antrl 725	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑎 ∈ ℝ)
26		nn0re 12477	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℝ)
27	26	ad2antll 726	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑏 ∈ ℝ)
28		simplrr 775	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℕ0)
29		simpll3 1211	. . . . . . . . . . . 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 774	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑎 ∈ ℕ0)
31		nn0z 12579	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℤ)
32		nn0z 12579	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℤ)
33		eluz 12832	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑏 ∈ (ℤ≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
34	31, 32, 33	syl2an 595	. . . . . . . . . . . . . 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 711	. . . . . . . . . . . 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 41904	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ (ℤ≥‘𝑎)) → (𝐹‘𝑎) ⊆ (𝐹‘𝑏))
38	29, 30, 36, 37	syl3anc 1368	. . . . . . . . . . 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 4172	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑎) ⊆ (𝐹‘𝑏) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑏))
40	38, 39	sylib 217	. . . . . . . . . 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 4032	. . . . . . . . . 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 4006	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)))
45	44	rspcev 3604	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
46	28, 42, 45	syl2anc 583	. . . . . . . 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 774	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℕ0)
48		simpll3 1211	. . . . . . . . . . . 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 775	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑏 ∈ ℕ0)
50		eluz 12832	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℤ ∧ 𝑎 ∈ ℤ) → (𝑎 ∈ (ℤ≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
51	32, 31, 50	syl2anr 596	. . . . . . . . . . . . . 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 711	. . . . . . . . . . . 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 41904	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑏 ∈ ℕ0 ∧ 𝑎 ∈ (ℤ≥‘𝑏)) → (𝐹‘𝑏) ⊆ (𝐹‘𝑎))
55	48, 49, 53, 54	syl3anc 1368	. . . . . . . . . . 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 4175	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑏) ⊆ (𝐹‘𝑎) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑎))
57	55, 56	sylib 217	. . . . . . . . . 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 4032	. . . . . . . . . 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 4006	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)))
62	61	rspcev 3604	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
63	47, 59, 62	syl2anc 583	. . . . . . . 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 11316	. . . . . . 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 3192	. . . . . 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 4148	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 ∪ 𝑧) = ((𝐹‘𝑎) ∪ 𝑧))
67	66	sseq1d 4005	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑎) → ((𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
68	67	rexbidv 3170	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑎) → (∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
69	68	ralbidv 3169	. . . . . . . . 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 7079	. . . . . . . 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 4149	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑏) → ((𝐹‘𝑎) ∪ 𝑧) = ((𝐹‘𝑎) ∪ (𝐹‘𝑏)))
72	71	sseq1d 4005	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
73	72	rexbidv 3170	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑏) → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
74	73	ralrn 7079	. . . . . . . . . 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 4000	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑐) → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
76	75	rexrn 7078	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℕ0 → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
77	76	ralbidv 3169	. . . . . . . . . 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 3169	. . . . . . . 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 18491	. . . . 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 1340	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (toInc‘ran 𝐹) ∈ Dirset)
85		isnacs3 41903	. . . . . . 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 1130	. . . . 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 2810	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → ((toInc‘𝑦) ∈ Dirset ↔ (toInc‘ran 𝐹) ∈ Dirset))
90		unieq 4910	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → ∪ 𝑦 = ∪ ran 𝐹)
91		id 22	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → 𝑦 = ran 𝐹)
92	90, 91	eleq12d 2819	. . . . . . 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 3602	. . . . 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 583	. . . 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 6942	. . . 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 231	. 2 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 (𝐹‘𝑦) = ∪ ran 𝐹)
100	10, 99	reximddv 3163	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 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2932  ∀wral 3053  ∃wrex 3062  Vcvv 3466   ∪ cun 3938   ⊆ wss 3940  ∅c0 4314  𝒫 cpw 4594  ∪ cuni 4899   class class class wbr 5138  ran crn 5667   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7401  ℝcr 11104  0cc0 11105  1c1 11106   + caddc 11108   ≤ cle 11245  ℕ₀cn0 12468  ℤcz 12554  ℤ_≥cuz 12818  Moorecmre 17524  Dirsetcdrs 18248  toInccipo 18481  NoeACScnacs 41895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17078  df-slot 17113  df-ndx 17125  df-base 17143  df-tset 17214  df-ple 17215  df-ocomp 17216  df-mre 17528  df-mrc 17529  df-acs 17531  df-proset 18249  df-drs 18250  df-poset 18267  df-ipo 18482  df-nacs 41896

This theorem is referenced by:  hbt  42327