Theorem sigaclci 34128

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 34123	. . . . . . . 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 1143	. . . . . 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 4572	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝑆)
6		ssrexv 4018	. . . . . . . . . . . 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 4031	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ {𝑦 ∣ ∃𝑧 ∈ 𝑆 𝑦 = (∪ 𝑆 ∖ 𝑧)})
9		isrnsigau 34123	. . . . . . . . . . . . 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 1143	. . . . . . . . . . 11 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑧 ∈ 𝑆 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆)
12		uniiunlem 4052	. . . . . . . . . . . 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 3963	. . . . . . . . 9 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆) → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ⊆ 𝑆)
16		abrexexg 7941	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ∈ V)
17		elpwg 4568	. . . . . . . . . . 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 1144	. . . . . . . . 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 32443	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝒫 𝑆 → {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ 𝐴)
25		domtr 8980	. . . . . . . . . 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 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 5112	. . . . . . . . 9 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → (𝑥 ≼ ω ↔ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} ≼ ω))
30		unieq 4884	. . . . . . . . . 10 ⊢ (𝑥 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)} → ∪ 𝑥 = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
31	30	eleq1d 2814	. . . . . . . . 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 3589	. . . . . . 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 4017	. . . . . . . 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 4996	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) ∈ 𝑆 → ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) = ∪ {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 = (∪ 𝑆 ∖ 𝑧)})
40		eleq1 2817	. . . . . . 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 4085	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → (∪ 𝑆 ∖ 𝑥) = (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)))
44	43	eleq1d 2814	. . . . . 6 ⊢ (𝑥 = ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧) → ((∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ↔ (∪ 𝑆 ∖ ∪ 𝑧 ∈ 𝐴 (∪ 𝑆 ∖ 𝑧)) ∈ 𝑆))
45	44	rspccv 3588	. . . . 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 4911	. . . . . . 7 ⊢ 𝑆 ⊆ 𝒫 ∪ 𝑆
51	5, 50	sstrdi 3961	. . . . . 6 ⊢ (𝐴 ∈ 𝒫 𝑆 → 𝐴 ⊆ 𝒫 ∪ 𝑆)
52		iundifdifd 32496	. . . . . 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 2817	. . . 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 1086   = wceq 1540   ∈ wcel 2109  {cab 2708   ≠ wne 2926  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∖ cdif 3913   ⊆ wss 3916  ∅c0 4298  𝒫 cpw 4565  ∪ cuni 4873  ∩ cint 4912  ∪ ciun 4957   class class class wbr 5109  ran crn 5641  ωcom 7844   ≼ cdom 8918  sigAlgebracsiga 34104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713  ax-ac2 10422

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-int 4913  df-iun 4959  df-iin 4960  df-br 5110  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-se 5594  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6276  df-ord 6337  df-on 6338  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522  df-riota 7346  df-ov 7392  df-oprab 7393  df-mpo 7394  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-er 8673  df-map 8803  df-en 8921  df-dom 8922  df-card 9898  df-acn 9901  df-ac 10075  df-siga 34105

This theorem is referenced by:  difelsiga  34129  sigapisys  34151