nacsfix - Mathbox for Stefan O'Rear

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Theorem nacsfix 41752

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 6923	. . . . 5 ⊢ (𝐹‘𝑧) ⊆ ∪ ran 𝐹
2		simplrr 774	. . . . 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 4034	. . . 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 1212	. . . . 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 773	. . . . 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 483	. . . . 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 41751	. . . . 5 ⊢ ((∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑦 ∈ ℕ₀ ∧ 𝑧 ∈ (ℤ_≥‘𝑦)) → (𝐹‘𝑦) ⊆ (𝐹‘𝑧))
8	4, 5, 6, 7	syl3anc 1369	. . . 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 3998	. . 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 3144	. 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 6723	. . . . . . . 8 ⊢ (𝐹:ℕ₀⟶𝐶 → ran 𝐹 ⊆ 𝐶)
12	11	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ⊆ 𝐶)
13		elpw2g 5343	. . . . . . . 8 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹 ⊆ 𝐶))
14	13	3ad2ant1 1131	. . . . . . 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 256	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ∈ 𝒫 𝐶)
16		elex 3491	. . . . . 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 6716	. . . . . . . 8 ⊢ (𝐹:ℕ₀⟶𝐶 → 𝐹 Fn ℕ₀)
19	18	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → 𝐹 Fn ℕ₀)
20		0nn0 12491	. . . . . . 7 ⊢ 0 ∈ ℕ₀
21		fnfvelrn 7081	. . . . . . 7 ⊢ ((𝐹 Fn ℕ₀ ∧ 0 ∈ ℕ₀) → (𝐹‘0) ∈ ran 𝐹)
22	19, 20, 21	sylancl 584	. . . . . 6 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (𝐹‘0) ∈ ran 𝐹)
23	22	ne0d 4334	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ≠ ∅)
24		nn0re 12485	. . . . . . . . 9 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 ∈ ℝ)
25	24	ad2antrl 724	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀)) → 𝑎 ∈ ℝ)
26		nn0re 12485	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℝ)
27	26	ad2antll 725	. . . . . . . 8 ⊢ (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀)) → 𝑏 ∈ ℝ)
28		simplrr 774	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀)) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ ℕ₀)
29		simpll3 1212	. . . . . . . . . . . 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 773	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀)) ∧ 𝑎 ≤ 𝑏) → 𝑎 ∈ ℕ₀)
31		nn0z 12587	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ℕ₀ → 𝑎 ∈ ℤ)
32		nn0z 12587	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ ℕ₀ → 𝑏 ∈ ℤ)
33		eluz 12840	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑏 ∈ (ℤ_≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
34	31, 32, 33	syl2an 594	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (𝑏 ∈ (ℤ_≥‘𝑎) ↔ 𝑎 ≤ 𝑏))
35	34	biimpar 476	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) ∧ 𝑎 ≤ 𝑏) → 𝑏 ∈ (ℤ_≥‘𝑎))
36	35	adantll 710	. . . . . . . . . . . 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 41751	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ (ℤ_≥‘𝑎)) → (𝐹‘𝑎) ⊆ (𝐹‘𝑏))
38	29, 30, 36, 37	syl3anc 1369	. . . . . . . . . . 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 4179	. . . . . . . . . . 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 4039	. . . . . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑏 → (𝐹‘𝑐) = (𝐹‘𝑏))
44	43	sseq2d 4013	. . . . . . . . . 10 ⊢ (𝑐 = 𝑏 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)))
45	44	rspcev 3611	. . . . . . . . 9 ⊢ ((𝑏 ∈ ℕ₀ ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑏)) → ∃𝑐 ∈ ℕ₀ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
46	28, 42, 45	syl2anc 582	. . . . . . . 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 773	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀)) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ ℕ₀)
48		simpll3 1212	. . . . . . . . . . . 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 774	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀)) ∧ 𝑏 ≤ 𝑎) → 𝑏 ∈ ℕ₀)
50		eluz 12840	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ℤ ∧ 𝑎 ∈ ℤ) → (𝑎 ∈ (ℤ_≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
51	32, 31, 50	syl2anr 595	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) → (𝑎 ∈ (ℤ_≥‘𝑏) ↔ 𝑏 ≤ 𝑎))
52	51	biimpar 476	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ ℕ₀ ∧ 𝑏 ∈ ℕ₀) ∧ 𝑏 ≤ 𝑎) → 𝑎 ∈ (ℤ_≥‘𝑏))
53	52	adantll 710	. . . . . . . . . . . 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 41751	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑏 ∈ ℕ₀ ∧ 𝑎 ∈ (ℤ_≥‘𝑏)) → (𝐹‘𝑏) ⊆ (𝐹‘𝑎))
55	48, 49, 53, 54	syl3anc 1369	. . . . . . . . . . 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 4182	. . . . . . . . . . 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 4039	. . . . . . . . . 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 6890	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑎 → (𝐹‘𝑐) = (𝐹‘𝑎))
61	60	sseq2d 4013	. . . . . . . . . 10 ⊢ (𝑐 = 𝑎 → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐) ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)))
62	61	rspcev 3611	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℕ₀ ∧ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑎)) → ∃𝑐 ∈ ℕ₀ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐))
63	47, 59, 62	syl2anc 582	. . . . . . . 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 11324	. . . . . . 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 3198	. . . . . 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 4155	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑎) → (𝑦 ∪ 𝑧) = ((𝐹‘𝑎) ∪ 𝑧))
67	66	sseq1d 4012	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑎) → ((𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
68	67	rexbidv 3176	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑎) → (∃𝑤 ∈ ran 𝐹(𝑦 ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤))
69	68	ralbidv 3175	. . . . . . . . 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 7088	. . . . . . . 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 4156	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑏) → ((𝐹‘𝑎) ∪ 𝑧) = ((𝐹‘𝑎) ∪ (𝐹‘𝑏)))
72	71	sseq1d 4012	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑏) → (((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
73	72	rexbidv 3176	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑏) → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤))
74	73	ralrn 7088	. . . . . . . . . 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 4007	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝐹‘𝑐) → (((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
76	75	rexrn 7087	. . . . . . . . . . 11 ⊢ (𝐹 Fn ℕ₀ → (∃𝑤 ∈ ran 𝐹((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ 𝑤 ↔ ∃𝑐 ∈ ℕ₀ ((𝐹‘𝑎) ∪ (𝐹‘𝑏)) ⊆ (𝐹‘𝑐)))
77	76	ralbidv 3175	. . . . . . . . . 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 278	. . . . . . . . 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 3175	. . . . . . . 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 278	. . . . . . 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 256	. . . . 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 1341	. . . 4 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (toInc‘ran 𝐹) ∈ Dirset)
85		isnacs3 41750	. . . . . . 7 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦)))
86	85	simprbi 495	. . . . . 6 ⊢ (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
87	86	3ad2ant1 1131	. . . . 5 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦))
88		fveq2 6890	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → (toInc‘𝑦) = (toInc‘ran 𝐹))
89	88	eleq1d 2816	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → ((toInc‘𝑦) ∈ Dirset ↔ (toInc‘ran 𝐹) ∈ Dirset))
90		unieq 4918	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → ∪ 𝑦 = ∪ ran 𝐹)
91		id 22	. . . . . . . 8 ⊢ (𝑦 = ran 𝐹 → 𝑦 = ran 𝐹)
92	90, 91	eleq12d 2825	. . . . . . 7 ⊢ (𝑦 = ran 𝐹 → (∪ 𝑦 ∈ 𝑦 ↔ ∪ ran 𝐹 ∈ ran 𝐹))
93	89, 92	imbi12d 343	. . . . . 6 ⊢ (𝑦 = ran 𝐹 → (((toInc‘𝑦) ∈ Dirset → ∪ 𝑦 ∈ 𝑦) ↔ ((toInc‘ran 𝐹) ∈ Dirset → ∪ ran 𝐹 ∈ ran 𝐹)))
94	93	rspcva 3609	. . . . 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 582	. . . 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 6951	. . . 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 3169	1 ⊢ ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ₀⟶𝐶 ∧ ∀𝑥 ∈ ℕ₀ (𝐹‘𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ₀ ∀𝑧 ∈ (ℤ_≥‘𝑦)(𝐹‘𝑧) = (𝐹‘𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 Vcvv 3472 ∪ cun 3945 ⊆ wss 3947 ∅c0 4321 𝒫 cpw 4601 ∪ cuni 4907 class class class wbr 5147 ran crn 5676 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 ≤ cle 11253 ℕ₀cn0 12476 ℤcz 12562 ℤ_≥cuz 12826 Moorecmre 17530 Dirsetcdrs 18251 toInccipo 18484 NoeACScnacs 41742

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-slot 17119 df-ndx 17131 df-base 17149 df-tset 17220 df-ple 17221 df-ocomp 17222 df-mre 17534 df-mrc 17535 df-acs 17537 df-proset 18252 df-drs 18253 df-poset 18270 df-ipo 18485 df-nacs 41743

This theorem is referenced by: hbt 42174

W3C validator