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Theorem nacsfix 36293
 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:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0𝑧 ∈ (ℤ𝑦)(𝐹𝑧) = (𝐹𝑦))
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.
StepHypRef Expression
1 fvssunirn 6127 . . . . 5 (𝐹𝑧) ⊆ ran 𝐹
2 simplrr 797 . . . . 5 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) ∧ 𝑧 ∈ (ℤ𝑦)) → (𝐹𝑦) = ran 𝐹)
31, 2syl5sseqr 3617 . . . 4 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) ∧ 𝑧 ∈ (ℤ𝑦)) → (𝐹𝑧) ⊆ (𝐹𝑦))
4 simpll3 1095 . . . . 5 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) ∧ 𝑧 ∈ (ℤ𝑦)) → ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)))
5 simplrl 796 . . . . 5 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) ∧ 𝑧 ∈ (ℤ𝑦)) → 𝑦 ∈ ℕ0)
6 simpr 476 . . . . 5 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) ∧ 𝑧 ∈ (ℤ𝑦)) → 𝑧 ∈ (ℤ𝑦))
7 incssnn0 36292 . . . . 5 ((∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑦 ∈ ℕ0𝑧 ∈ (ℤ𝑦)) → (𝐹𝑦) ⊆ (𝐹𝑧))
84, 5, 6, 7syl3anc 1318 . . . 4 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) ∧ 𝑧 ∈ (ℤ𝑦)) → (𝐹𝑦) ⊆ (𝐹𝑧))
93, 8eqssd 3585 . . 3 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) ∧ 𝑧 ∈ (ℤ𝑦)) → (𝐹𝑧) = (𝐹𝑦))
109ralrimiva 2949 . 2 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑦 ∈ ℕ0 ∧ (𝐹𝑦) = ran 𝐹)) → ∀𝑧 ∈ (ℤ𝑦)(𝐹𝑧) = (𝐹𝑦))
11 frn 5966 . . . . . . . 8 (𝐹:ℕ0𝐶 → ran 𝐹𝐶)
12113ad2ant2 1076 . . . . . . 7 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹𝐶)
13 elpw2g 4754 . . . . . . . 8 (𝐶 ∈ (NoeACS‘𝑋) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹𝐶))
14133ad2ant1 1075 . . . . . . 7 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (ran 𝐹 ∈ 𝒫 𝐶 ↔ ran 𝐹𝐶))
1512, 14mpbird 246 . . . . . 6 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ∈ 𝒫 𝐶)
16 elex 3185 . . . . . 6 (ran 𝐹 ∈ 𝒫 𝐶 → ran 𝐹 ∈ V)
1715, 16syl 17 . . . . 5 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ∈ V)
18 ffn 5958 . . . . . . . 8 (𝐹:ℕ0𝐶𝐹 Fn ℕ0)
19183ad2ant2 1076 . . . . . . 7 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → 𝐹 Fn ℕ0)
20 0nn0 11184 . . . . . . 7 0 ∈ ℕ0
21 fnfvelrn 6264 . . . . . . 7 ((𝐹 Fn ℕ0 ∧ 0 ∈ ℕ0) → (𝐹‘0) ∈ ran 𝐹)
2219, 20, 21sylancl 693 . . . . . 6 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (𝐹‘0) ∈ ran 𝐹)
23 ne0i 3880 . . . . . 6 ((𝐹‘0) ∈ ran 𝐹 → ran 𝐹 ≠ ∅)
2422, 23syl 17 . . . . 5 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ≠ ∅)
25 nn0re 11178 . . . . . . . . 9 (𝑎 ∈ ℕ0𝑎 ∈ ℝ)
2625ad2antrl 760 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) → 𝑎 ∈ ℝ)
27 nn0re 11178 . . . . . . . . 9 (𝑏 ∈ ℕ0𝑏 ∈ ℝ)
2827ad2antll 761 . . . . . . . 8 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) → 𝑏 ∈ ℝ)
29 simplrr 797 . . . . . . . . 9 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → 𝑏 ∈ ℕ0)
30 simpll3 1095 . . . . . . . . . . . 12 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)))
31 simplrl 796 . . . . . . . . . . . 12 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → 𝑎 ∈ ℕ0)
32 nn0z 11277 . . . . . . . . . . . . . . 15 (𝑎 ∈ ℕ0𝑎 ∈ ℤ)
33 nn0z 11277 . . . . . . . . . . . . . . 15 (𝑏 ∈ ℕ0𝑏 ∈ ℤ)
34 eluz 11577 . . . . . . . . . . . . . . 15 ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℤ) → (𝑏 ∈ (ℤ𝑎) ↔ 𝑎𝑏))
3532, 33, 34syl2an 493 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑏 ∈ (ℤ𝑎) ↔ 𝑎𝑏))
3635biimpar 501 . . . . . . . . . . . . 13 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ 𝑎𝑏) → 𝑏 ∈ (ℤ𝑎))
3736adantll 746 . . . . . . . . . . . 12 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → 𝑏 ∈ (ℤ𝑎))
38 incssnn0 36292 . . . . . . . . . . . 12 ((∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑎 ∈ ℕ0𝑏 ∈ (ℤ𝑎)) → (𝐹𝑎) ⊆ (𝐹𝑏))
3930, 31, 37, 38syl3anc 1318 . . . . . . . . . . 11 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → (𝐹𝑎) ⊆ (𝐹𝑏))
40 ssequn1 3745 . . . . . . . . . . 11 ((𝐹𝑎) ⊆ (𝐹𝑏) ↔ ((𝐹𝑎) ∪ (𝐹𝑏)) = (𝐹𝑏))
4139, 40sylib 207 . . . . . . . . . 10 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → ((𝐹𝑎) ∪ (𝐹𝑏)) = (𝐹𝑏))
42 eqimss 3620 . . . . . . . . . 10 (((𝐹𝑎) ∪ (𝐹𝑏)) = (𝐹𝑏) → ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑏))
4341, 42syl 17 . . . . . . . . 9 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑏))
44 fveq2 6103 . . . . . . . . . . 11 (𝑐 = 𝑏 → (𝐹𝑐) = (𝐹𝑏))
4544sseq2d 3596 . . . . . . . . . 10 (𝑐 = 𝑏 → (((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐) ↔ ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑏)))
4645rspcev 3282 . . . . . . . . 9 ((𝑏 ∈ ℕ0 ∧ ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑏)) → ∃𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐))
4729, 43, 46syl2anc 691 . . . . . . . 8 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑎𝑏) → ∃𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐))
48 simplrl 796 . . . . . . . . 9 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → 𝑎 ∈ ℕ0)
49 simpll3 1095 . . . . . . . . . . . 12 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)))
50 simplrr 797 . . . . . . . . . . . 12 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → 𝑏 ∈ ℕ0)
51 eluz 11577 . . . . . . . . . . . . . . 15 ((𝑏 ∈ ℤ ∧ 𝑎 ∈ ℤ) → (𝑎 ∈ (ℤ𝑏) ↔ 𝑏𝑎))
5233, 32, 51syl2anr 494 . . . . . . . . . . . . . 14 ((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) → (𝑎 ∈ (ℤ𝑏) ↔ 𝑏𝑎))
5352biimpar 501 . . . . . . . . . . . . 13 (((𝑎 ∈ ℕ0𝑏 ∈ ℕ0) ∧ 𝑏𝑎) → 𝑎 ∈ (ℤ𝑏))
5453adantll 746 . . . . . . . . . . . 12 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → 𝑎 ∈ (ℤ𝑏))
55 incssnn0 36292 . . . . . . . . . . . 12 ((∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1)) ∧ 𝑏 ∈ ℕ0𝑎 ∈ (ℤ𝑏)) → (𝐹𝑏) ⊆ (𝐹𝑎))
5649, 50, 54, 55syl3anc 1318 . . . . . . . . . . 11 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → (𝐹𝑏) ⊆ (𝐹𝑎))
57 ssequn2 3748 . . . . . . . . . . 11 ((𝐹𝑏) ⊆ (𝐹𝑎) ↔ ((𝐹𝑎) ∪ (𝐹𝑏)) = (𝐹𝑎))
5856, 57sylib 207 . . . . . . . . . 10 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → ((𝐹𝑎) ∪ (𝐹𝑏)) = (𝐹𝑎))
59 eqimss 3620 . . . . . . . . . 10 (((𝐹𝑎) ∪ (𝐹𝑏)) = (𝐹𝑎) → ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑎))
6058, 59syl 17 . . . . . . . . 9 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑎))
61 fveq2 6103 . . . . . . . . . . 11 (𝑐 = 𝑎 → (𝐹𝑐) = (𝐹𝑎))
6261sseq2d 3596 . . . . . . . . . 10 (𝑐 = 𝑎 → (((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐) ↔ ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑎)))
6362rspcev 3282 . . . . . . . . 9 ((𝑎 ∈ ℕ0 ∧ ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑎)) → ∃𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐))
6448, 60, 63syl2anc 691 . . . . . . . 8 ((((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) ∧ 𝑏𝑎) → ∃𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐))
6526, 28, 47, 64lecasei 10022 . . . . . . 7 (((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) ∧ (𝑎 ∈ ℕ0𝑏 ∈ ℕ0)) → ∃𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐))
6665ralrimivva 2954 . . . . . 6 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐))
67 uneq1 3722 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑎) → (𝑦𝑧) = ((𝐹𝑎) ∪ 𝑧))
6867sseq1d 3595 . . . . . . . . . . 11 (𝑦 = (𝐹𝑎) → ((𝑦𝑧) ⊆ 𝑤 ↔ ((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤))
6968rexbidv 3034 . . . . . . . . . 10 (𝑦 = (𝐹𝑎) → (∃𝑤 ∈ ran 𝐹(𝑦𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤))
7069ralbidv 2969 . . . . . . . . 9 (𝑦 = (𝐹𝑎) → (∀𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹(𝑦𝑧) ⊆ 𝑤 ↔ ∀𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤))
7170ralrn 6270 . . . . . . . 8 (𝐹 Fn ℕ0 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹(𝑦𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤))
72 uneq2 3723 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑏) → ((𝐹𝑎) ∪ 𝑧) = ((𝐹𝑎) ∪ (𝐹𝑏)))
7372sseq1d 3595 . . . . . . . . . . . 12 (𝑧 = (𝐹𝑏) → (((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ 𝑤))
7473rexbidv 3034 . . . . . . . . . . 11 (𝑧 = (𝐹𝑏) → (∃𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∃𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ 𝑤))
7574ralrn 6270 . . . . . . . . . 10 (𝐹 Fn ℕ0 → (∀𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑏 ∈ ℕ0𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ 𝑤))
76 sseq2 3590 . . . . . . . . . . . 12 (𝑤 = (𝐹𝑐) → (((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ 𝑤 ↔ ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐)))
7776rexrn 6269 . . . . . . . . . . 11 (𝐹 Fn ℕ0 → (∃𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ 𝑤 ↔ ∃𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐)))
7877ralbidv 2969 . . . . . . . . . 10 (𝐹 Fn ℕ0 → (∀𝑏 ∈ ℕ0𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ 𝑤 ↔ ∀𝑏 ∈ ℕ0𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐)))
7975, 78bitrd 267 . . . . . . . . 9 (𝐹 Fn ℕ0 → (∀𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑏 ∈ ℕ0𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐)))
8079ralbidv 2969 . . . . . . . 8 (𝐹 Fn ℕ0 → (∀𝑎 ∈ ℕ0𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹((𝐹𝑎) ∪ 𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐)))
8171, 80bitrd 267 . . . . . . 7 (𝐹 Fn ℕ0 → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹(𝑦𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐)))
8219, 81syl 17 . . . . . 6 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹(𝑦𝑧) ⊆ 𝑤 ↔ ∀𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑐 ∈ ℕ0 ((𝐹𝑎) ∪ (𝐹𝑏)) ⊆ (𝐹𝑐)))
8366, 82mpbird 246 . . . . 5 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹(𝑦𝑧) ⊆ 𝑤)
84 isipodrs 16984 . . . . 5 ((toInc‘ran 𝐹) ∈ Dirset ↔ (ran 𝐹 ∈ V ∧ ran 𝐹 ≠ ∅ ∧ ∀𝑦 ∈ ran 𝐹𝑧 ∈ ran 𝐹𝑤 ∈ ran 𝐹(𝑦𝑧) ⊆ 𝑤))
8517, 24, 83, 84syl3anbrc 1239 . . . 4 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → (toInc‘ran 𝐹) ∈ Dirset)
86 isnacs3 36291 . . . . . . 7 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → 𝑦𝑦)))
8786simprbi 479 . . . . . 6 (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → 𝑦𝑦))
88873ad2ant1 1075 . . . . 5 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → 𝑦𝑦))
89 fveq2 6103 . . . . . . . 8 (𝑦 = ran 𝐹 → (toInc‘𝑦) = (toInc‘ran 𝐹))
9089eleq1d 2672 . . . . . . 7 (𝑦 = ran 𝐹 → ((toInc‘𝑦) ∈ Dirset ↔ (toInc‘ran 𝐹) ∈ Dirset))
91 unieq 4380 . . . . . . . 8 (𝑦 = ran 𝐹 𝑦 = ran 𝐹)
92 id 22 . . . . . . . 8 (𝑦 = ran 𝐹𝑦 = ran 𝐹)
9391, 92eleq12d 2682 . . . . . . 7 (𝑦 = ran 𝐹 → ( 𝑦𝑦 ran 𝐹 ∈ ran 𝐹))
9490, 93imbi12d 333 . . . . . 6 (𝑦 = ran 𝐹 → (((toInc‘𝑦) ∈ Dirset → 𝑦𝑦) ↔ ((toInc‘ran 𝐹) ∈ Dirset → ran 𝐹 ∈ ran 𝐹)))
9594rspcva 3280 . . . . 5 ((ran 𝐹 ∈ 𝒫 𝐶 ∧ ∀𝑦 ∈ 𝒫 𝐶((toInc‘𝑦) ∈ Dirset → 𝑦𝑦)) → ((toInc‘ran 𝐹) ∈ Dirset → ran 𝐹 ∈ ran 𝐹))
9615, 88, 95syl2anc 691 . . . 4 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ((toInc‘ran 𝐹) ∈ Dirset → ran 𝐹 ∈ ran 𝐹))
9785, 96mpd 15 . . 3 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ran 𝐹 ∈ ran 𝐹)
98 fvelrnb 6153 . . . 4 (𝐹 Fn ℕ0 → ( ran 𝐹 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ℕ0 (𝐹𝑦) = ran 𝐹))
9919, 98syl 17 . . 3 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ( ran 𝐹 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ℕ0 (𝐹𝑦) = ran 𝐹))
10097, 99mpbid 221 . 2 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0 (𝐹𝑦) = ran 𝐹)
10110, 100reximddv 3001 1 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝐹:ℕ0𝐶 ∧ ∀𝑥 ∈ ℕ0 (𝐹𝑥) ⊆ (𝐹‘(𝑥 + 1))) → ∃𝑦 ∈ ℕ0𝑧 ∈ (ℤ𝑦)(𝐹𝑧) = (𝐹𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   ≤ cle 9954  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  Moorecmre 16065  Dirsetcdrs 16750  toInccipo 16974  NoeACScnacs 36283 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-tset 15787  df-ple 15788  df-ocomp 15790  df-mre 16069  df-mrc 16070  df-acs 16072  df-preset 16751  df-drs 16752  df-poset 16769  df-ipo 16975  df-nacs 36284 This theorem is referenced by:  hbt  36719
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