Theorem sigaclci 33118

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 33113	. . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
2	1	simprd 496	. . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
3	2	simp2d 1143	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
4	3	adantr 481	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
5		elpwi 4608	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝑆)
6		ssrexv 4050	. . . . . . . . . . . 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 4059	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)})
9		isrnsigau 33113	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → ∪ 𝑧 ∈ 𝑆))))
10	9	simprd 496	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → ∪ 𝑧 ∈ 𝑆)))
11	10	simp2d 1143	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
12		uniiunlem 4083	. . . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆)
15	8, 14	sylan9ssr 3995	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆)
16		abrexexg 7943	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ V)
17		elpwg 4604	. . . . . . . . . . 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 482	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆)
21	2	simp3d 1144	. . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
22	21	adantr 481	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
23	20, 22	jca 512	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
24		abrexdom2jm 31732	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ 𝐴)
25		domtr 8999	. . . . . . . . . 10 ⊢ (({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ 𝐴 ∧ 𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω)
26	24, 25	sylan 580	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω)
27	26	ex 413	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 𝑆 → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
28	27	adantl 482	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
29		breq1 5150	. . . . . . . . 9 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
30		unieq 4918	. . . . . . . . . 10 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → ∪ 𝑥 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
31	30	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → (∪ 𝑥 ∈ 𝑆 ↔ ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝑆))
32	29, 31	imbi12d 344	. . . . . . . 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 3610	. . . . . . 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 482	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ⊆ 𝑆)
36	11	adantr 481	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
37		ssralv 4049	. . . . . . . 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 5032	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 → ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
40		eleq1 2821	. . . . . . 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 258	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆))
43		difeq2 4115	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)))
44	43	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
45	44	rspccv 3609	. . . . 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 492	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
48	47	imp 407	. 2 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆)
49		simpr 485	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ∈ 𝒫 𝑆)
50		pwuni 4948	. . . . . . 7 ⊢ 𝑆 ⊆ 𝒫 ∪ 𝑆
51	5, 50	sstrdi 3993	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝒫 ∪ 𝑆)
52		iundifdifd 31780	. . . . . 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 491	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧))))
55		eleq1 2821	. . . 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 407	. 2 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → (∩ 𝐴 ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
58	48, 57	mpbird 256	1 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ∩ 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  {cab 2709   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3944   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  ∪ cuni 4907  ∩ cint 4949  ∪ ciun 4996   class class class wbr 5147  ran crn 5676  ωcom 7851   ≼ cdom 8933  sigAlgebracsiga 33094

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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-ac2 10454

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-iin 4999  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-se 5631  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 6297  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-card 9930  df-acn 9933  df-ac 10107  df-siga 33095

This theorem is referenced by:  difelsiga  33119  sigapisys  33141