isacs2 - Metamath Proof Explorer

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Theorem isacs2 16314

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 16312	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))))
2		iunss 4561	. . . . . . . . 9 ⊢ (∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)
3		ffun 6048	. . . . . . . . . . 11 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → Fun 𝑓)
4		funiunfv 6506	. . . . . . . . . . 11 ⊢ (Fun 𝑓 → ∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) = ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)))
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → ∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) = ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)))
6	5	sseq1d 3632	. . . . . . . . 9 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → (∪ 𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡))
7	2, 6	syl5rbbr 275	. . . . . . . 8 ⊢ (𝑓:𝒫 𝑋⟶𝒫 𝑋 → (∪ (𝑓 “ (𝒫 𝑡 ∩ Fin)) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡))
8	7	bibi2d 332	. . . . . . 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 2986	. . . . . 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 669	. . . . 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 1774	. . . 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 790	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝐶 ∈ (Moore‘𝑋))
13		inss1 3833	. . . . . . . . . . . . . . . 16 ⊢ (𝒫 𝑠 ∩ Fin) ⊆ 𝒫 𝑠
14	13	sseli 3599	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ∈ 𝒫 𝑠)
15		elpwi 4168	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝒫 𝑠 → 𝑦 ⊆ 𝑠)
16	14, 15	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ⊆ 𝑠)
17	16	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑦 ⊆ 𝑠)
18		simplr 792	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → 𝑠 ∈ 𝐶)
19		isacs2.f	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (mrCls‘𝐶)
20	19	mrcsscl 16280	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ⊆ 𝑠 ∧ 𝑠 ∈ 𝐶) → (𝐹‘𝑦) ⊆ 𝑠)
21	12, 17, 18, 20	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) ∧ 𝑦 ∈ (𝒫 𝑠 ∩ Fin)) → (𝐹‘𝑦) ⊆ 𝑠)
22	21	ralrimiva 2966	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)
23	22	adantlr 751	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑠 ∈ 𝐶) → ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)
24	23	adantllr 755	. . . . . . . . 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)
25		simplll 798	. . . . . . . . . . . . . . . . . 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 ∈ (Moore‘𝑋))
26	16	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((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)
27		elpwi 4168	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ 𝒫 𝑋 → 𝑠 ⊆ 𝑋)
28	27	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((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)
29	26, 28	sstrd 3613	. . . . . . . . . . . . . . . . . 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 ⊆ 𝑋)
30	25, 19, 29	mrcssidd 16285	. . . . . . . . . . . . . . . . 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))
31		vex 3203	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑦 ∈ V
32	31	elpw 4164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ 𝒫 (𝐹‘𝑦) ↔ 𝑦 ⊆ (𝐹‘𝑦))
33	30, 32	sylibr 224	. . . . . . . . . . . . . . . 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))
34		inss2 3834	. . . . . . . . . . . . . . . . . 18 ⊢ (𝒫 𝑠 ∩ Fin) ⊆ Fin
35	34	sseli 3599	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (𝒫 𝑠 ∩ Fin) → 𝑦 ∈ Fin)
36	35	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)
37	33, 36	elind 3798	. . . . . . . . . . . . . . 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))
38	19	mrccl 16271	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑦 ⊆ 𝑋) → (𝐹‘𝑦) ∈ 𝐶)
39	25, 29, 38	syl2anc 693	. . . . . . . . . . . . . . . 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)
40		mresspw 16252	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋)
41	40	ad3antrrr 766	. . . . . . . . . . . . . . . . . 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)
42	41, 39	sseldd 3604	. . . . . . . . . . . . . . . . 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)
43		simprr 796	. . . . . . . . . . . . . . . . . 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))
44	43	ad2antrr 762	. . . . . . . . . . . . . . . . 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))
45		eleq1 2689	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐹‘𝑦) → (𝑡 ∈ 𝐶 ↔ (𝐹‘𝑦) ∈ 𝐶))
46		pweq 4161	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑡 = (𝐹‘𝑦) → 𝒫 𝑡 = 𝒫 (𝐹‘𝑦))
47	46	ineq1d 3813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐹‘𝑦) → (𝒫 𝑡 ∩ Fin) = (𝒫 (𝐹‘𝑦) ∩ Fin))
48		sseq2 3627	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑡 = (𝐹‘𝑦) → ((𝑓‘𝑧) ⊆ 𝑡 ↔ (𝑓‘𝑧) ⊆ (𝐹‘𝑦)))
49	47, 48	raleqbidv 3152	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 = (𝐹‘𝑦) → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦)))
50	45, 49	bibi12d 335	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = (𝐹‘𝑦) → ((𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ ((𝐹‘𝑦) ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦))))
51	50	rspcva 3307	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹‘𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)) → ((𝐹‘𝑦) ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦)))
52	42, 44, 51	syl2anc 693	. . . . . . . . . . . . . . . 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)))
53	39, 52	mpbid 222	. . . . . . . . . . . . . . 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))
54		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = 𝑦 → (𝑓‘𝑧) = (𝑓‘𝑦))
55	54	sseq1d 3632	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑦 → ((𝑓‘𝑧) ⊆ (𝐹‘𝑦) ↔ (𝑓‘𝑦) ⊆ (𝐹‘𝑦)))
56	55	rspcva 3307	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin) ∧ ∀𝑧 ∈ (𝒫 (𝐹‘𝑦) ∩ Fin)(𝑓‘𝑧) ⊆ (𝐹‘𝑦)) → (𝑓‘𝑦) ⊆ (𝐹‘𝑦))
57	37, 53, 56	syl2anc 693	. . . . . . . . . . . . . 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))
58		sstr2 3610	. . . . . . . . . . . . . 14 ⊢ ((𝑓‘𝑦) ⊆ (𝐹‘𝑦) → ((𝐹‘𝑦) ⊆ 𝑠 → (𝑓‘𝑦) ⊆ 𝑠))
59	57, 58	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))
60	59	ralimdva 2962	. . . . . . . . . . . 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))
61	60	imp 445	. . . . . . . . . . 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)
62		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑧 → (𝑓‘𝑦) = (𝑓‘𝑧))
63	62	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → ((𝑓‘𝑦) ⊆ 𝑠 ↔ (𝑓‘𝑧) ⊆ 𝑠))
64	63	cbvralv 3171	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑦) ⊆ 𝑠 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠)
65	61, 64	sylib 208	. . . . . . . . . 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)
66		simplr 792	. . . . . . . . . . 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)
67	43	ad2antrr 762	. . . . . . . . . . 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))
68		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (𝑡 ∈ 𝐶 ↔ 𝑠 ∈ 𝐶))
69		pweq 4161	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑠 → 𝒫 𝑡 = 𝒫 𝑠)
70	69	ineq1d 3813	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → (𝒫 𝑡 ∩ Fin) = (𝒫 𝑠 ∩ Fin))
71		sseq2 3627	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑠 → ((𝑓‘𝑧) ⊆ 𝑡 ↔ (𝑓‘𝑧) ⊆ 𝑠))
72	70, 71	raleqbidv 3152	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠))
73	68, 72	bibi12d 335	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → ((𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ (𝑠 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠)))
74	73	rspcva 3307	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ 𝒫 𝑋 ∧ ∀𝑡 ∈ 𝒫 𝑋(𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡)) → (𝑠 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑠 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑠))
75	66, 67, 74	syl2anc 693	. . . . . . . . . 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))
76	65, 75	mpbird 247	. . . . . . . . 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)
77	24, 76	impbida 877	. . . . . . . 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))
78	77	ralrimiva 2966	. . . . . . 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))
79	78	ex 450	. . . . . 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)))
80	79	exlimdv 1861	. . . . 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)))
81	19	mrcf 16269	. . . . . . . 8 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
82	81, 40	fssd 6057	. . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝒫 𝑋)
83		fvex 6201	. . . . . . . . 9 ⊢ (mrCls‘𝐶) ∈ V
84	19, 83	eqeltri 2697	. . . . . . . 8 ⊢ 𝐹 ∈ V
85		feq1 6026	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓:𝒫 𝑋⟶𝒫 𝑋 ↔ 𝐹:𝒫 𝑋⟶𝒫 𝑋))
86		fveq1 6190	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑧) = (𝐹‘𝑧))
87	86	sseq1d 3632	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑧) ⊆ 𝑡 ↔ (𝐹‘𝑧) ⊆ 𝑡))
88	87	ralbidv 2986	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑧) ⊆ 𝑡))
89		fveq2 6191	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑦 → (𝐹‘𝑧) = (𝐹‘𝑦))
90	89	sseq1d 3632	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑦 → ((𝐹‘𝑧) ⊆ 𝑡 ↔ (𝐹‘𝑦) ⊆ 𝑡))
91	90	cbvralv 3171	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑧) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡)
92	88, 91	syl6bb 276	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡))
93	92	bibi2d 332	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ((𝑡 ∈ 𝐶 ↔ ∀𝑧 ∈ (𝒫 𝑡 ∩ Fin)(𝑓‘𝑧) ⊆ 𝑡) ↔ (𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡)))
94	93	ralbidv 2986	. . . . . . . . . 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)))
95		sseq2 3627	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → ((𝐹‘𝑦) ⊆ 𝑡 ↔ (𝐹‘𝑦) ⊆ 𝑠))
96	70, 95	raleqbidv 3152	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑠 → (∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
97	68, 96	bibi12d 335	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑠 → ((𝑡 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑡 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑡) ↔ (𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
98	97	cbvralv 3171	. . . . . . . . . 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))
99	94, 98	syl6bb 276	. . . . . . . . 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)))
100	85, 99	anbi12d 747	. . . . . . . 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))))
101	84, 100	spcev 3300	. . . . . . 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)))
102	82, 101	sylan 488	. . . . . 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)))
103	102	ex 450	. . . . 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))))
104	80, 103	impbid 202	. . . 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)))
105	11, 104	syl5bb 272	. . 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)))
106	105	pm5.32i 669	. 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)))
107	1, 106	bitri 264	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 196 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 𝒫 cpw 4158 ∪ cuni 4436 ∪ ciun 4520 “ cima 5117 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 Fincfn 7955 Moorecmre 16242 mrClscmrc 16243 ACScacs 16245

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-mre 16246 df-mrc 16247 df-acs 16249

This theorem is referenced by: acsfiel 16315 isacs5 17172

W3C validator