Theorem sigaclci 31386

Description: A sigma-algebra is closed under countable intersections. Deduction version. (Contributed by Thierry Arnoux, 19-Sep-2016.)

Assertion

Ref	Expression
sigaclci	⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ∩ 𝐴 ∈ 𝑆)

Proof of Theorem sigaclci

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isrnsigau 31381	. . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
2	1	simprd 498	. . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
3	2	simp2d 1139	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
4	3	adantr 483	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
5		elpwi 4550	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝑆)
6		ssrexv 4033	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ 𝑆 → (∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧) → ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)))
7	5, 6	syl 17	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 𝑆 → (∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧) → ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)))
8	7	ss2abdv 4043	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)})
9		isrnsigau 31381	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → ∪ 𝑧 ∈ 𝑆))))
10	9	simprd 498	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → ∪ 𝑧 ∈ 𝑆)))
11	10	simp2d 1139	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
12		uniiunlem 4060	. . . . . . . . . . . 12 ⊢ (∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 → (∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ↔ {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆))
13	11, 12	syl 17	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ↔ {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆))
14	11, 13	mpbid 234	. . . . . . . . . 10 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆)
15	8, 14	sylan9ssr 3980	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆)
16		abrexexg 7656	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ V)
17		elpwg 4544	. . . . . . . . . . 11 ⊢ ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ V → ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆))
18	16, 17	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆))
19	18	adantl 484	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆 ↔ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆))
20	15, 19	mpbird 259	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆)
21	2	simp3d 1140	. . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
22	21	adantr 483	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
23	20, 22	jca 514	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
24		abrexdom2jm 30262	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ 𝐴)
25		domtr 8556	. . . . . . . . . 10 ⊢ (({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ 𝐴 ∧ 𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω)
26	24, 25	sylan 582	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω)
27	26	ex 415	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 𝑆 → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
28	27	adantl 484	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
29		breq1 5061	. . . . . . . . 9 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
30		unieq 4839	. . . . . . . . . 10 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → ∪ 𝑥 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
31	30	eleq1d 2897	. . . . . . . . 9 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → (∪ 𝑥 ∈ 𝑆 ↔ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝑆))
32	29, 31	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) ↔ ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝑆)))
33	32	rspcva 3620	. . . . . . 7 ⊢ (({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)) → ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝑆))
34	23, 28, 33	sylsyld 61	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝑆))
35	5	adantl 484	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ⊆ 𝑆)
36	11	adantr 483	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
37		ssralv 4032	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝑆 → (∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 → ∀𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆))
38	35, 36, 37	sylc 65	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
39		dfiun2g 4947	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 → ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
40		eleq1 2900	. . . . . . 7 ⊢ (∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → (∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ↔ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝑆))
41	38, 39, 40	3syl 18	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ↔ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝑆))
42	34, 41	sylibrd 261	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆))
43		difeq2 4092	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)))
44	43	eleq1d 2897	. . . . . 6 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
45	44	rspccv 3619	. . . . 5 ⊢ (∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 → (∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
46	4, 42, 45	sylsyld 61	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
47	46	adantrd 494	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
48	47	imp 409	. 2 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆)
49		simpr 487	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ∈ 𝒫 𝑆)
50		pwuni 4867	. . . . . . 7 ⊢ 𝑆 ⊆ 𝒫 ∪ 𝑆
51	5, 50	sstrdi 3978	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝒫 ∪ 𝑆)
52		iundifdifd 30307	. . . . . 6 ⊢ (𝐴 ⊆ 𝒫 ∪ 𝑆 → (𝐴 ≠ ∅ → ∩ 𝐴 = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧))))
53	49, 51, 52	3syl 18	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≠ ∅ → ∩ 𝐴 = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧))))
54	53	adantld 493	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧))))
55		eleq1 2900	. . . 4 ⊢ (∩ 𝐴 = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) → (∩ 𝐴 ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
56	54, 55	syl6 35	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∩ 𝐴 ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆)))
57	56	imp 409	. 2 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → (∩ 𝐴 ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
58	48, 57	mpbird 259	1 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ∩ 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∖ cdif 3932   ⊆ wss 3935  ∅c0 4290  𝒫 cpw 4538  ∪ cuni 4831  ∩ cint 4868  ∪ ciun 4911   class class class wbr 5058  ran crn 5550  ωcom 7574   ≼ cdom 8501  sigAlgebracsiga 31362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-ac2 9879

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-card 9362  df-acn 9365  df-ac 9536  df-siga 31363

This theorem is referenced by:  difelsiga  31387  sigapisys  31409