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Theorem sectffval 16617
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 6333 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 issect.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2823 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fvexd 6346 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V)
7 fveq2 6333 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 issect.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
97, 8syl6eqr 2823 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
10 simpr 471 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
1110oveqd 6813 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1211eleq2d 2836 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑓 ∈ (𝑥𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
1310oveqd 6813 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑦𝑥) = (𝑦𝐻𝑥))
1413eleq2d 2836 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔 ∈ (𝑦𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥)))
1512, 14anbi12d 616 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ↔ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥))))
16 simpl 468 . . . . . . . . . . . . 13 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6337 . . . . . . . . . . . 12 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 issect.o . . . . . . . . . . . 12 · = (comp‘𝐶)
1917, 18syl6eqr 2823 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = · )
2019oveqd 6813 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑦· 𝑥))
2120oveqd 6813 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓))
2216fveq2d 6337 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = (Id‘𝐶))
23 issect.i . . . . . . . . . . 11 1 = (Id‘𝐶)
2422, 23syl6eqr 2823 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = 1 )
2524fveq1d 6335 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
2621, 25eqeq12d 2786 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥) ↔ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥)))
2715, 26anbi12d 616 . . . . . . 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 3625 . . . . . 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 4851 . . . . 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, 29mpt2eq123dv 6868 . . . 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 16614 . . . 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 6345 . . . . 5 𝐵 ∈ V
3332, 32mpt2ex 7401 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}) ∈ V
3430, 31, 33fvmpt 6426 . . 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 2817 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 382   = wceq 1631  wcel 2145  Vcvv 3351  [wsbc 3587  cop 4323  {copab 4847  cfv 6030  (class class class)co 6796  cmpt2 6798  Basecbs 16064  Hom chom 16160  compcco 16161  Catccat 16532  Idccid 16533  Sectcsect 16611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-sect 16614
This theorem is referenced by:  sectfval  16618
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