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Theorem sectffval 17592
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 6839 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 issect.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4eqtr4di 2795 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fvexd 6854 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V)
7 fveq2 6839 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 issect.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
97, 8eqtr4di 2795 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
10 simpr 485 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
1110oveqd 7368 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1211eleq2d 2823 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑓 ∈ (𝑥𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
1310oveqd 7368 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑦𝑥) = (𝑦𝐻𝑥))
1413eleq2d 2823 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔 ∈ (𝑦𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥)))
1512, 14anbi12d 631 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ↔ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥))))
16 simpl 483 . . . . . . . . . . . . 13 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6843 . . . . . . . . . . . 12 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 issect.o . . . . . . . . . . . 12 · = (comp‘𝐶)
1917, 18eqtr4di 2795 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = · )
2019oveqd 7368 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑦· 𝑥))
2120oveqd 7368 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓))
2216fveq2d 6843 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = (Id‘𝐶))
23 issect.i . . . . . . . . . . 11 1 = (Id‘𝐶)
2422, 23eqtr4di 2795 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = 1 )
2524fveq1d 6841 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
2621, 25eqeq12d 2753 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥) ↔ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥)))
2715, 26anbi12d 631 . . . . . . 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 3784 . . . . . 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 5169 . . . . 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 7426 . . . 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 17589 . . . 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 6853 . . . . 5 𝐵 ∈ V
3332, 32mpoex 8004 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}) ∈ V
3430, 31, 33fvmpt 6945 . . 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, 35eqtrid 2789 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 396   = wceq 1541  wcel 2106  Vcvv 3443  [wsbc 3737  cop 4590  {copab 5165  cfv 6493  (class class class)co 7351  cmpo 7353  Basecbs 17042  Hom chom 17103  compcco 17104  Catccat 17503  Idccid 17504  Sectcsect 17586
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-sect 17589
This theorem is referenced by:  sectfval  17593
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