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Theorem tendoset 39225
Description: The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom π‘Š. (Contributed by NM, 8-Jun-2013.)
Hypotheses
Ref Expression
tendoset.l ≀ = (leβ€˜πΎ)
tendoset.h 𝐻 = (LHypβ€˜πΎ)
tendoset.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
tendoset.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
tendoset.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
tendoset ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐸 = {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
Distinct variable groups:   𝑓,𝑠,𝑔,𝐾   𝑇,𝑓,𝑔,𝑠   π‘Š,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 𝑀 is distinct from all other variables.
StepHypRef Expression
1 tendoset.e . 2 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
2 tendoset.l . . . . 5 ≀ = (leβ€˜πΎ)
3 tendoset.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
42, 3tendofset 39224 . . . 4 (𝐾 ∈ 𝑉 β†’ (TEndoβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))}))
54fveq1d 6845 . . 3 (𝐾 ∈ 𝑉 β†’ ((TEndoβ€˜πΎ)β€˜π‘Š) = ((𝑀 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))})β€˜π‘Š))
6 fveq2 6843 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
76, 6feq23d 6664 . . . . . . 7 (𝑀 = π‘Š β†’ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ↔ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š)))
86raleqdv 3314 . . . . . . . 8 (𝑀 = π‘Š β†’ (βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ↔ βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”))))
96, 8raleqbidv 3320 . . . . . . 7 (𝑀 = π‘Š β†’ (βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ↔ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”))))
10 fveq2 6843 . . . . . . . . . . 11 (𝑀 = π‘Š β†’ ((trLβ€˜πΎ)β€˜π‘€) = ((trLβ€˜πΎ)β€˜π‘Š))
11 tendoset.r . . . . . . . . . . 11 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
1210, 11eqtr4di 2795 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ((trLβ€˜πΎ)β€˜π‘€) = 𝑅)
1312fveq1d 6845 . . . . . . . . 9 (𝑀 = π‘Š β†’ (((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) = (π‘…β€˜(π‘ β€˜π‘“)))
1412fveq1d 6845 . . . . . . . . 9 (𝑀 = π‘Š β†’ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) = (π‘…β€˜π‘“))
1513, 14breq12d 5119 . . . . . . . 8 (𝑀 = π‘Š β†’ ((((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ↔ (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“)))
166, 15raleqbidv 3320 . . . . . . 7 (𝑀 = π‘Š β†’ (βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) ↔ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“)))
177, 9, 163anbi123d 1437 . . . . . 6 (𝑀 = π‘Š β†’ ((𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“)) ↔ (𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))))
1817abbidv 2806 . . . . 5 (𝑀 = π‘Š β†’ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))} = {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
19 eqid 2737 . . . . 5 (𝑀 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))}) = (𝑀 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))})
20 fvex 6856 . . . . . . . 8 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
2120, 20mapval 8778 . . . . . . 7 (((LTrnβ€˜πΎ)β€˜π‘Š) ↑m ((LTrnβ€˜πΎ)β€˜π‘Š)) = {𝑠 ∣ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š)}
22 ovex 7391 . . . . . . 7 (((LTrnβ€˜πΎ)β€˜π‘Š) ↑m ((LTrnβ€˜πΎ)β€˜π‘Š)) ∈ V
2321, 22eqeltrri 2835 . . . . . 6 {𝑠 ∣ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š)} ∈ V
24 simp1 1137 . . . . . . 7 ((𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“)) β†’ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š))
2524ss2abi 4024 . . . . . 6 {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))} βŠ† {𝑠 ∣ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š)}
2623, 25ssexi 5280 . . . . 5 {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))} ∈ V
2718, 19, 26fvmpt 6949 . . . 4 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))})β€˜π‘Š) = {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
28 tendoset.t . . . . . . 7 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
2928, 28feq23i 6663 . . . . . 6 (𝑠:π‘‡βŸΆπ‘‡ ↔ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š))
3028raleqi 3312 . . . . . . 7 (βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ↔ βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)))
3128, 30raleqbii 3316 . . . . . 6 (βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ↔ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)))
3228raleqi 3312 . . . . . 6 (βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“) ↔ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))
3329, 31, 323anbi123i 1156 . . . . 5 ((𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“)) ↔ (𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“)))
3433abbii 2807 . . . 4 {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))} = {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘Š)(π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))}
3527, 34eqtr4di 2795 . . 3 (π‘Š ∈ 𝐻 β†’ ((𝑀 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrnβ€˜πΎ)β€˜π‘€)⟢((LTrnβ€˜πΎ)β€˜π‘€) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)βˆ€π‘” ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ ((LTrnβ€˜πΎ)β€˜π‘€)(((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) ≀ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“))})β€˜π‘Š) = {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
365, 35sylan9eq 2797 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ ((TEndoβ€˜πΎ)β€˜π‘Š) = {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
371, 36eqtrid 2789 1 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐸 = {𝑠 ∣ (𝑠:π‘‡βŸΆπ‘‡ ∧ βˆ€π‘“ ∈ 𝑇 βˆ€π‘” ∈ 𝑇 (π‘ β€˜(𝑓 ∘ 𝑔)) = ((π‘ β€˜π‘“) ∘ (π‘ β€˜π‘”)) ∧ βˆ€π‘“ ∈ 𝑇 (π‘…β€˜(π‘ β€˜π‘“)) ≀ (π‘…β€˜π‘“))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  Vcvv 3446   class class class wbr 5106   ↦ cmpt 5189   ∘ ccom 5638  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8766  lecple 17141  LHypclh 38450  LTrncltrn 38567  trLctrl 38624  TEndoctendo 39218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8768  df-tendo 39221
This theorem is referenced by:  istendo  39226
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