Theorem tendoset 40716

Description: The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom 𝑊. (Contributed by NM, 8-Jun-2013.)

Hypotheses

Ref	Expression
tendoset.l	⊢ ≤ = (le‘𝐾)
tendoset.h	⊢ 𝐻 = (LHyp‘𝐾)
tendoset.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
tendoset.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
tendoset.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendoset	⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})

Distinct variable groups:   𝑓,𝑠,𝑔,𝐾   𝑇,𝑓,𝑔,𝑠   𝑊,𝑠,𝑓,𝑔

Allowed substitution hints:   𝑅(𝑓,𝑔,𝑠)   𝐸(𝑓,𝑔,𝑠)   𝐻(𝑓,𝑔,𝑠)   ≤ (𝑓,𝑔,𝑠)   𝑉(𝑓,𝑔,𝑠)

Proof of Theorem tendoset

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		tendoset.e	. 2 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
2		tendoset.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		tendoset.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4	2, 3	tendofset 40715	. . . 4 ⊢ (𝐾 ∈ 𝑉 → (TEndo‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))}))
5	4	fveq1d 6922	. . 3 ⊢ (𝐾 ∈ 𝑉 → ((TEndo‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))})‘𝑊))
6		fveq2 6920	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((LTrn‘𝐾)‘𝑤) = ((LTrn‘𝐾)‘𝑊))
7	6, 6	feq23d 6742	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ↔ 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)))
8	6	raleqdv 3334	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔))))
9	6, 8	raleqbidv 3354	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔))))
10		fveq2 6920	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = ((trL‘𝐾)‘𝑊))
11		tendoset.r	. . . . . . . . . . 11 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
12	10, 11	eqtr4di 2798	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((trL‘𝐾)‘𝑤) = 𝑅)
13	12	fveq1d 6922	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) = (𝑅‘(𝑠‘𝑓)))
14	12	fveq1d 6922	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((trL‘𝐾)‘𝑤)‘𝑓) = (𝑅‘𝑓))
15	13, 14	breq12d 5179	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓) ↔ (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓)))
16	6, 15	raleqbidv 3354	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓)))
17	7, 9, 16	3anbi123d 1436	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓)) ↔ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))))
18	17	abbidv 2811	. . . . 5 ⊢ (𝑤 = 𝑊 → {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))} = {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})
19		eqid 2740	. . . . 5 ⊢ (𝑤 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))}) = (𝑤 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))})
20		fvex 6933	. . . . . . . 8 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
21	20, 20	mapval 8896	. . . . . . 7 ⊢ (((LTrn‘𝐾)‘𝑊) ↑m ((LTrn‘𝐾)‘𝑊)) = {𝑠 ∣ 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)}
22		ovex 7481	. . . . . . 7 ⊢ (((LTrn‘𝐾)‘𝑊) ↑m ((LTrn‘𝐾)‘𝑊)) ∈ V
23	21, 22	eqeltrri 2841	. . . . . 6 ⊢ {𝑠 ∣ 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)} ∈ V
24		simp1 1136	. . . . . . 7 ⊢ ((𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓)) → 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
25	24	ss2abi 4090	. . . . . 6 ⊢ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))} ⊆ {𝑠 ∣ 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊)}
26	23, 25	ssexi 5340	. . . . 5 ⊢ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))} ∈ V
27	18, 19, 26	fvmpt 7029	. . . 4 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))})‘𝑊) = {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})
28		tendoset.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
29	28, 28	feq23i 6741	. . . . . 6 ⊢ (𝑠:𝑇⟶𝑇 ↔ 𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊))
30	28	raleqi 3332	. . . . . . 7 ⊢ (∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ ∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)))
31	28, 30	raleqbii 3352	. . . . . 6 ⊢ (∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)))
32	28	raleqi 3332	. . . . . 6 ⊢ (∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓) ↔ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))
33	29, 31, 32	3anbi123i 1155	. . . . 5 ⊢ ((𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓)) ↔ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓)))
34	33	abbii 2812	. . . 4 ⊢ {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))} = {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑊)⟶((LTrn‘𝐾)‘𝑊) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))}
35	27, 34	eqtr4di 2798	. . 3 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))})‘𝑊) = {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})
36	5, 35	sylan9eq 2800	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ((TEndo‘𝐾)‘𝑊) = {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})
37	1, 36	eqtrid 2792	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488   class class class wbr 5166   ↦ cmpt 5249   ∘ ccom 5704  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884  lecple 17318  LHypclh 39941  LTrncltrn 40058  trLctrl 40115  TEndoctendo 40709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-tendo 40712

This theorem is referenced by:  istendo  40717