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Theorem sectffval 17012
Description: Value of the section operation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
issect.b 𝐵 = (Base‘𝐶)
issect.h 𝐻 = (Hom ‘𝐶)
issect.o · = (comp‘𝐶)
issect.i 1 = (Id‘𝐶)
issect.s 𝑆 = (Sect‘𝐶)
issect.c (𝜑𝐶 ∈ Cat)
issect.x (𝜑𝑋𝐵)
issect.y (𝜑𝑌𝐵)
Assertion
Ref Expression
sectffval (𝜑𝑆 = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦, 1   𝑥,𝐵,𝑦   𝐶,𝑓,𝑔,𝑥,𝑦   𝜑,𝑓,𝑔,𝑥,𝑦   𝑓,𝐻,𝑔,𝑥,𝑦   · ,𝑓,𝑔,𝑥,𝑦   𝑓,𝑋,𝑔,𝑥,𝑦   𝑓,𝑌,𝑔,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑓,𝑔)   𝑆(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem sectffval
Dummy variables 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issect.s . 2 𝑆 = (Sect‘𝐶)
2 issect.c . . 3 (𝜑𝐶 ∈ Cat)
3 fveq2 6645 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 issect.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2851 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fvexd 6660 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V)
7 fveq2 6645 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 issect.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
97, 8eqtr4di 2851 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
10 simpr 488 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
1110oveqd 7152 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1211eleq2d 2875 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑓 ∈ (𝑥𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
1310oveqd 7152 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑦𝑥) = (𝑦𝐻𝑥))
1413eleq2d 2875 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔 ∈ (𝑦𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥)))
1512, 14anbi12d 633 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ↔ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥))))
16 simpl 486 . . . . . . . . . . . . 13 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6649 . . . . . . . . . . . 12 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 issect.o . . . . . . . . . . . 12 · = (comp‘𝐶)
1917, 18eqtr4di 2851 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = · )
2019oveqd 7152 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑦· 𝑥))
2120oveqd 7152 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓))
2216fveq2d 6649 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = (Id‘𝐶))
23 issect.i . . . . . . . . . . 11 1 = (Id‘𝐶)
2422, 23eqtr4di 2851 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = 1 )
2524fveq1d 6647 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
2621, 25eqeq12d 2814 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥) ↔ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥)))
2715, 26anbi12d 633 . . . . . . 7 ((𝑐 = 𝐶 = 𝐻) → (((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥)) ↔ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))))
286, 9, 27sbcied2 3763 . . . . . 6 (𝑐 = 𝐶 → ([(Hom ‘𝑐) / ]((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥)) ↔ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))))
2928opabbidv 5096 . . . . 5 (𝑐 = 𝐶 → {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ]((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))})
305, 5, 29mpoeq123dv 7208 . . . 4 (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ]((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}) = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
31 df-sect 17009 . . . 4 Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ]((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}))
324fvexi 6659 . . . . 5 𝐵 ∈ V
3332, 32mpoex 7760 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}) ∈ V
3430, 31, 33fvmpt 6745 . . 3 (𝐶 ∈ Cat → (Sect‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
352, 34syl 17 . 2 (𝜑 → (Sect‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
361, 35syl5eq 2845 1 (𝜑𝑆 = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  [wsbc 3720  cop 4531  {copab 5092  cfv 6324  (class class class)co 7135  cmpo 7137  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Idccid 16928  Sectcsect 17006
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-sect 17009
This theorem is referenced by:  sectfval  17013
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