Theorem nacsfix 43161

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 6858	. . . . 5 ⊢ (𝐹‘𝑧) ⊆ ∪ ran 𝐹
2		simplrr 783	. . . . 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 3958	. . . 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 1221	. . . . 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 782	. . . . 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 485	. . . . 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 43160	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑦 ∈ ℕ0 ∧ 𝑧 ∈ (ℤ≥‘𝑦)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑧))
8	4, 5, 6, 7	syl3anc 1379	. . . 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 3932	. . 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 3131	. 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 6662	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → ran 𝐹 ⊆ 𝐶)
12	11	3ad2ant2 1140	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ⊆ 𝐶)
13		elpw2g 5261	. . . . . . . 8 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
14	13	3ad2ant1 1139	. . . . . . 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 258	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ∈ 𝒫 𝐶)
16		elex 3452	. . . . . 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 6655	. . . . . . . 8 ⊢ (𝐹:ℕ0⟶𝐶 → 𝐹 Fn ℕ0)
19	18	3ad2ant2 1140	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → 𝐹 Fn ℕ0)
20		0nn0 12443	. . . . . . 7 ⊢ 0 ∈ ℕ0
21		fnfvelrn 7021	. . . . . . 7 ⊢ ((𝐹 Fn ℕ0 ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ ran 𝐹)
22	19, 20, 21	sylancl 592	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (𝐹‘0) ∈ ran 𝐹)
23	22	ne0d 4270	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ≠ ∅)
24		nn0re 12437	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℝ)
25	24	ad2antrl 734	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑎 ∈ ℝ)
26		nn0re 12437	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℝ)
27	26	ad2antll 735	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) → 𝑏 ∈ ℝ)
28		simplrr 783	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℕ0)
29		simpll3 1221	. . . . . . . . . . . 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 782	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑎 ≤ 𝑏) → 𝑎 ∈ ℕ0)
31		nn0z 12539	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ0 → 𝑎 ∈ ℤ)
32		nn0z 12539	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ0 → 𝑏 ∈ ℤ)
33		eluz 12793	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑏 ∈ (ℤ≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
34	31, 32, 33	syl2an 602	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑏 ∈ (ℤ≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
35	34	biimpar 478	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ (ℤ≥‘𝑎))
36	35	adantll 720	. . . . . . . . . . . 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 43160	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑎 ∈ ℕ0 ∧ 𝑏 ∈ (ℤ≥‘𝑎)) → (𝐹‘𝑎) ⊆ (𝐹‘𝑏))
38	29, 30, 36, 37	syl3anc 1379	. . . . . . . . . . 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 4115	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑎) ⊆ (𝐹‘𝑏) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑏))
40	38, 39	sylib 219	. . . . . . . . . 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 3973	. . . . . . . . . 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 6827	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (𝐹‘𝑐) = (𝐹‘𝑏))
44	43	sseq2d 3947	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)))
45	44	rspcev 3560	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
46	28, 42, 45	syl2anc 590	. . . . . . . 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 782	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℕ0)
48		simpll3 1221	. . . . . . . . . . . 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 783	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0)) ∧ 𝑏 ≤ 𝑎) → 𝑏 ∈ ℕ0)
50		eluz 12793	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℤ ∧ 𝑎 ∈ ℤ) → (𝑎 ∈ (ℤ≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
51	32, 31, 50	syl2anr 603	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) → (𝑎 ∈ (ℤ≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
52	51	biimpar 478	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ (ℤ≥‘𝑏))
53	52	adantll 720	. . . . . . . . . . . 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 43160	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑏 ∈ ℕ0 ∧ 𝑎 ∈ (ℤ≥‘𝑏)) → (𝐹‘𝑏) ⊆ (𝐹‘𝑎))
55	48, 49, 53, 54	syl3anc 1379	. . . . . . . . . . 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 4118	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑏) ⊆ (𝐹‘𝑎) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) = (𝐹‘𝑎))
57	55, 56	sylib 219	. . . . . . . . . 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 3973	. . . . . . . . . 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 6827	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → (𝐹‘𝑐) = (𝐹‘𝑎))
61	60	sseq2d 3947	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)))
62	61	rspcev 3560	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ0 ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)) → ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
63	47, 59, 62	syl2anc 590	. . . . . . . 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 11243	. . . . . . 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 3182	. . . . . 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 4091	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 ∪ 𝑧) = ((𝐹‘𝑎) ∪ 𝑧))
67	66	sseq1d 3946	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑎) → ((𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
68	67	rexbidv 3163	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑎) → (∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
69	68	ralbidv 3162	. . . . . . . . 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 7029	. . . . . . . 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 4092	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑏) → ((𝐹‘𝑎) ∪ 𝑧) = ((𝐹‘𝑎) ∪ (𝐹‘𝑏)))
72	71	sseq1d 3946	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
73	72	rexbidv 3163	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑏) → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
74	73	ralrn 7029	. . . . . . . . . 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 3941	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑐) → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
76	75	rexrn 7028	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℕ0 → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ∃𝑐 ∈ ℕ0 ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
77	76	ralbidv 3162	. . . . . . . . . 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 280	. . . . . . . . 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 3162	. . . . . . . 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 280	. . . . . . 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 258	. . . . 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 18494	. . . . 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 1350	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (toInc‘ran 𝐹) ∈ Dirset)
85		isnacs3 43159	. . . . . . 7 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦)))
86	85	simprbi 498	. . . . . 6 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
87	86	3ad2ant1 1139	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
88		fveq2 6827	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → (toInc‘𝑦) = (toInc‘ran 𝐹))
89	88	eleq1d 2824	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → ((toInc‘𝑦) ∈ Dirset ↔ (toInc‘ran 𝐹) ∈ Dirset))
90		unieq 4849	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → ∪ 𝑦 = ∪ ran 𝐹)
91		id 22	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → 𝑦 = ran 𝐹)
92	90, 91	eleq12d 2833	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ ran 𝐹 ∈ ran 𝐹))
93	89, 92	imbi12d 345	. . . . . 6 ⊢ (𝑦 = ran 𝐹 → (((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦) ↔ ((toInc‘ran 𝐹) ∈ Dirset → ∪ ran 𝐹 ∈ ran 𝐹)))
94	93	rspcva 3558	. . . . 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 590	. . . 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 6887	. . . 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 233	. 2 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0⟶𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 (𝐹‘𝑦) = ∪ ran 𝐹)
100	10, 99	reximddv 3155	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 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ∪ cuni 4838   class class class wbr 5072  ran crn 5619   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  ℝcr 11028  0cc0 11029  1c1 11030   + caddc 11032   ≤ cle 11171  ℕ₀cn0 12428  ℤcz 12515  ℤ_≥cuz 12779  Moorecmre 17535  Dirsetcdrs 18250  toInccipo 18484  NoeACScnacs 43151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-3 12236  df-4 12237  df-5 12238  df-6 12239  df-7 12240  df-8 12241  df-9 12242  df-n0 12429  df-z 12516  df-dec 12636  df-uz 12780  df-fz 13453  df-struct 17108  df-slot 17143  df-ndx 17155  df-base 17171  df-tset 17230  df-ple 17231  df-ocomp 17232  df-mre 17539  df-mrc 17540  df-acs 17542  df-proset 18251  df-drs 18252  df-poset 18270  df-ipo 18485  df-nacs 43152

This theorem is referenced by:  hbt  43575