Theorem sigaclci 34276

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 34271	. . . . . . . 8 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
2	1	simprd 495	. . . . . . 7 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
3	2	simp2d 1144	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
4	3	adantr 480	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆)
5		elpwi 4548	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝑆)
6		ssrexv 3991	. . . . . . . . . . . 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 4005	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)})
9		isrnsigau 34271	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → ∪ 𝑧 ∈ 𝑆))))
10	9	simprd 495	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 ∧ ∀𝑧 ∈ 𝒫 𝑆(𝑧 ≼ ω → ∪ 𝑧 ∈ 𝑆)))
11	10	simp2d 1144	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
12		uniiunlem 4027	. . . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆)
15	8, 14	sylan9ssr 3936	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆)
16		abrexexg 7914	. . . . . . . . . . 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 481	. . . . . . . . 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 257	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆)
21	2	simp3d 1145	. . . . . . . . 9 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
22	21	adantr 480	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
23	20, 22	jca 511	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ 𝒫 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
24		abrexdom2jm 32578	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ 𝐴)
25		domtr 8954	. . . . . . . . . 10 ⊢ (({𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ 𝐴 ∧ 𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω)
26	24, 25	sylan 581	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω)
27	26	ex 412	. . . . . . . 8 ⊢ (𝐴 ∈ 𝒫 𝑆 → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
28	27	adantl 481	. . . . . . 7 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
29		breq1 5088	. . . . . . . . 9 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
30		unieq 4861	. . . . . . . . . 10 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → ∪ 𝑥 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
31	30	eleq1d 2821	. . . . . . . . 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 3562	. . . . . . 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 481	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ⊆ 𝑆)
36	11	adantr 480	. . . . . . . 8 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
37		ssralv 3990	. . . . . . . 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 4972	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 → ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
40		eleq1 2824	. . . . . . 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 259	. . . . 5 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → (𝐴 ≼ ω → ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆))
43		difeq2 4060	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)))
44	43	eleq1d 2821	. . . . . 6 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
45	44	rspccv 3561	. . . . 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 491	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
48	47	imp 406	. 2 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆)
49		simpr 484	. . . . . 6 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → 𝐴 ∈ 𝒫 𝑆)
50		pwuni 4888	. . . . . . 7 ⊢ 𝑆 ⊆ 𝒫 ∪ 𝑆
51	5, 50	sstrdi 3934	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝒫 ∪ 𝑆)
52		iundifdifd 32631	. . . . . 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 490	. . . 4 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → ((𝐴 ≼ ω ∧ 𝐴 ≠ ∅) → ∩ 𝐴 = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧))))
55		eleq1 2824	. . . 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 406	. 2 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → (∩ 𝐴 ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
58	48, 57	mpbird 257	1 ⊢ (((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) ∧ (𝐴 ≼ ω ∧ 𝐴 ≠ ∅)) → ∩ 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2714   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ∖ cdif 3886   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  ∪ cuni 4850  ∩ cint 4889  ∪ ciun 4933   class class class wbr 5085  ran crn 5632  ωcom 7817   ≼ cdom 8891  sigAlgebracsiga 34252

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-card 9863  df-acn 9866  df-ac 10038  df-siga 34253

This theorem is referenced by:  difelsiga  34277  sigapisys  34299