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Theorem tendoset 40272
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 40271 . . . 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 6904 . . 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 6902 . . . . . . . 8 (𝑀 = π‘Š β†’ ((LTrnβ€˜πΎ)β€˜π‘€) = ((LTrnβ€˜πΎ)β€˜π‘Š))
76, 6feq23d 6722 . . . . . . 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 3323 . . . . . . . 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 3340 . . . . . . 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 6902 . . . . . . . . . . 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 2786 . . . . . . . . . 10 (𝑀 = π‘Š β†’ ((trLβ€˜πΎ)β€˜π‘€) = 𝑅)
1312fveq1d 6904 . . . . . . . . 9 (𝑀 = π‘Š β†’ (((trLβ€˜πΎ)β€˜π‘€)β€˜(π‘ β€˜π‘“)) = (π‘…β€˜(π‘ β€˜π‘“)))
1412fveq1d 6904 . . . . . . . . 9 (𝑀 = π‘Š β†’ (((trLβ€˜πΎ)β€˜π‘€)β€˜π‘“) = (π‘…β€˜π‘“))
1513, 14breq12d 5165 . . . . . . . 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 3340 . . . . . . 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 1432 . . . . . 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 2797 . . . . 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 2728 . . . . 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 6915 . . . . . . . 8 ((LTrnβ€˜πΎ)β€˜π‘Š) ∈ V
2120, 20mapval 8865 . . . . . . 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 7459 . . . . . . 7 (((LTrnβ€˜πΎ)β€˜π‘Š) ↑m ((LTrnβ€˜πΎ)β€˜π‘Š)) ∈ V
2321, 22eqeltrri 2826 . . . . . 6 {𝑠 ∣ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š)} ∈ V
24 simp1 1133 . . . . . . 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 4063 . . . . . 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 5326 . . . . 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 7010 . . . 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 6721 . . . . . 6 (𝑠:π‘‡βŸΆπ‘‡ ↔ 𝑠:((LTrnβ€˜πΎ)β€˜π‘Š)⟢((LTrnβ€˜πΎ)β€˜π‘Š))
3028raleqi 3321 . . . . . . 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 3336 . . . . . 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 3321 . . . . . 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 1152 . . . . 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 2798 . . . 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 2786 . . 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 2788 . 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 2780 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2705  βˆ€wral 3058  Vcvv 3473   class class class wbr 5152   ↦ cmpt 5235   ∘ ccom 5686  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ↑m cmap 8853  lecple 17249  LHypclh 39497  LTrncltrn 39614  trLctrl 39671  TEndoctendo 40265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8855  df-tendo 40268
This theorem is referenced by:  istendo  40273
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