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Theorem isacs2 17533

Description: In the definition of an algebraic closure system, we may always take the operation being closed over as the Moore closure. (Contributed by Stefan O'Rear, 2-Apr-2015.)

**Hypothesis**
Ref	Expression
isacs2.f	⊢ 𝐹 = (mrCls‘𝐶)

**Assertion**
Ref	Expression
isacs2	⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))

Distinct variable groups: 𝐶,𝑠,𝑦 𝐹,𝑠,𝑦 𝑋,𝑠,𝑦

Proof of Theorem isacs2 Dummy variables 𝑓 𝑡 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isacs 17531	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
2		ffun 6671	. . . . . . . . . . 11 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → Fun 𝑓)
3		funiunfv 7195	. . . . . . . . . . 11 ⊢ (Fun 𝑓 → ∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) = ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)))
4	2, 3	syl 17	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → ∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) = ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)))
5	4	sseq1d 3975	. . . . . . . . 9 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → (∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
6		iunss 5005	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)
7	5, 6	bitr3di 285	. . . . . . . 8 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → (∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))
8	7	bibi2d 342	. . . . . . 7 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → ((𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ (𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)))
9	8	ralbidv 3174	. . . . . 6 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → (∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡) ↔ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)))
10	9	pm5.32i 575	. . . . 5 ⊢ ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)))
11	10	exbii 1850	. . . 4 ⊢ (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)))
12		simpll 765	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
13		elinel1 4155	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ∈ 𝒫 𝑠)
14	13	elpwid 4569	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ⊆ 𝑠)
15	14	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ⊆ 𝑠)
16		simplr 767	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑠 ∈ 𝐶)
17		isacs2.f	. . . . . . . . . . . . 13 ⊢ 𝐹 = (mrCls‘𝐶)
18	17	mrcsscl 17500	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ⊆ 𝑠 ∧ 𝑠 ∈ 𝐶) → (𝐹‘𝑦) ⊆ 𝑠)
19	12, 15, 16, 18	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹‘𝑦) ⊆ 𝑠)
20	19	ralrimiva 3143	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)
21	20	ad4ant14 750	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑠 ∈ 𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)
22		fveq2 6842	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → (𝑓‘𝑧) = (𝑓‘𝑦))
23	22	sseq1d 3975	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → ((𝑓‘𝑧) ⊆ (𝐹‘𝑦) ↔ (𝑓‘𝑦) ⊆ (𝐹‘𝑦)))
24		simplll 773	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
25	14	adantl 482	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ⊆ 𝑠)
26		elpwi 4567	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋)
27	26	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑠 ⊆ 𝑋)
28	25, 27	sstrd 3954	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ⊆ 𝑋)
29	17	mrccl 17491	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ⊆ 𝑋) → (𝐹‘𝑦) ∈ 𝐶)
30	24, 28, 29	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹‘𝑦) ∈ 𝐶)
31		eleq1 2825	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐹‘𝑦) → (𝑡 ∈ 𝐶 ↔ (𝐹‘𝑦) ∈ 𝐶))
32		pweq 4574	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐹‘𝑦) → 𝒫 𝑡 = 𝒫 (𝐹‘𝑦))
33	32	ineq1d 4171	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐹‘𝑦) → (𝒫 𝑡 ∩ Fin) = (𝒫 (𝐹‘𝑦) ∩ Fin))
34		sseq2 3970	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐹‘𝑦) → ((𝑓‘𝑧) ⊆ 𝑡 ↔ (𝑓‘𝑧) ⊆ (𝐹‘𝑦)))
35	33, 34	raleqbidv 3319	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐹‘𝑦) → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦)))
36	31, 35	bibi12d 345	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = (𝐹‘𝑦) → ((𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ ((𝐹‘𝑦) ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦))))
37		simprr 771	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) → ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))
38	37	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))
39		mresspw 17472	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
40	39	ad3antrrr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ⊆ 𝒫 𝑋)
41	40, 30	sseldd 3945	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹‘𝑦) ∈ 𝒫 𝑋)
42	36, 38, 41	rspcdva 3582	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ((𝐹‘𝑦) ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦)))
43	30, 42	mpbid 231	. . . . . . . . . . . . . . 15 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦))
44	24, 17, 28	mrcssidd 17505	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ⊆ (𝐹‘𝑦))
45		vex 3449	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
46	45	elpw 4564	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝒫 (𝐹‘𝑦) ↔ 𝑦 ⊆ (𝐹‘𝑦))
47	44, 46	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ∈ 𝒫 (𝐹‘𝑦))
48		elinel2 4156	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ∈ Fin)
49	48	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ∈ Fin)
50	47, 49	elind 4154	. . . . . . . . . . . . . . 15 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin))
51	23, 43, 50	rspcdva 3582	. . . . . . . . . . . . . 14 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝑓‘𝑦) ⊆ (𝐹‘𝑦))
52		sstr2 3951	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑦) ⊆ (𝐹‘𝑦) → ((𝐹‘𝑦) ⊆ 𝑠 → (𝑓‘𝑦) ⊆ 𝑠))
53	51, 52	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → ((𝐹‘𝑦) ⊆ 𝑠 → (𝑓‘𝑦) ⊆ 𝑠))
54	53	ralimdva 3164	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠 → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑦) ⊆ 𝑠))
55	54	imp 407	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑦) ⊆ 𝑠)
56		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑓‘𝑦) = (𝑓‘𝑧))
57	56	sseq1d 3975	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑓‘𝑦) ⊆ 𝑠 ↔ (𝑓‘𝑧) ⊆ 𝑠))
58	57	cbvralvw 3225	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑦) ⊆ 𝑠 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠)
59	55, 58	sylib 217	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) → ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠)
60		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ 𝐶 ↔ 𝑠 ∈ 𝐶))
61		pweq 4574	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → 𝒫 𝑡 = 𝒫 𝑠)
62	61	ineq1d 4171	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (𝒫 𝑡 ∩ Fin) = (𝒫 𝑠 ∩ Fin))
63		sseq2 3970	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → ((𝑓‘𝑧) ⊆ 𝑡 ↔ (𝑓‘𝑧) ⊆ 𝑠))
64	62, 63	raleqbidv 3319	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠))
65	60, 64	bibi12d 345	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → ((𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ (𝑠 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠)))
66	37	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) → ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))
67		simplr 767	. . . . . . . . . . 11 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) → 𝑠 ∈ 𝒫 𝑋)
68	65, 66, 67	rspcdva 3582	. . . . . . . . . 10 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) → (𝑠 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠))
69	59, 68	mpbird 256	. . . . . . . . 9 ⊢ ((((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) → 𝑠 ∈ 𝐶)
70	21, 69	impbida 799	. . . . . . . 8 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) ∧ 𝑠 ∈ 𝒫 𝑋) → (𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
71	70	ralrimiva 3143	. . . . . . 7 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
72	71	ex 413	. . . . . 6 ⊢ (𝐶 ∈ (Moore‘𝑋) → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
73	72	exlimdv 1936	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
74	17	mrcf 17489	. . . . . . . 8 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
75	74, 39	fssd 6686	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
76	17	fvexi 6856	. . . . . . . 8 ⊢ 𝐹 ∈ V
77		feq1 6649	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋 ↔ 𝐹:𝒫 𝑋⟶𝒫 𝑋))
78		fveq1 6841	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧))
79	78	sseq1d 3975	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑧) ⊆ 𝑡 ↔ (𝐹‘𝑧) ⊆ 𝑡))
80	79	ralbidv 3174	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑧) ⊆ 𝑡))
81		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
82	81	sseq1d 3975	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) ⊆ 𝑡 ↔ (𝐹‘𝑦) ⊆ 𝑡))
83	82	cbvralvw 3225	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑧) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡)
84	80, 83	bitrdi 286	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡))
85	84	bibi2d 342	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ (𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡)))
86	85	ralbidv 3174	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡)))
87		sseq2 3970	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → ((𝐹‘𝑦) ⊆ 𝑡 ↔ (𝐹‘𝑦) ⊆ 𝑠))
88	62, 87	raleqbidv 3319	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → (∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
89	60, 88	bibi12d 345	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → ((𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡) ↔ (𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
90	89	cbvralvw 3225	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
91	86, 90	bitrdi 286	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
92	77, 91	anbi12d 631	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)) ↔ (𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))))
93	76, 92	spcev 3565	. . . . . . 7 ⊢ ((𝐹:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)))
94	75, 93	sylan 580	. . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)))
95	94	ex 413	. . . . 5 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) → ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))))
96	73, 95	impbid 211	. . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
97	11, 96	bitrid 282	. . 3 ⊢ (𝐶 ∈ (Moore‘𝑋) → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡)) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
98	97	pm5.32i 575	. 2 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
99	1, 98	bitri 274	1 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3064 ∩ cin 3909 ⊆ wss 3910 𝒫 cpw 4560 ∪ cuni 4865 ∪ ciun 4954 “ cima 5636 Fun wfun 6490 ⟶wf 6492 ‘cfv 6496 Fincfn 8883 Moorecmre 17462 mrClscmrc 17463 ACScacs 17465

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-mre 17466 df-mrc 17467 df-acs 17469

This theorem is referenced by: acsfiel 17534 isacs5 18437

W3C validator