MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sectffval Structured version   Visualization version   GIF version

Theorem sectffval 16391
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 6178 . . . . . 6 (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
4 issect.b . . . . . 6 𝐵 = (Base‘𝐶)
53, 4syl6eqr 2672 . . . . 5 (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
6 fvexd 6190 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) ∈ V)
7 fveq2 6178 . . . . . . . 8 (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
8 issect.h . . . . . . . 8 𝐻 = (Hom ‘𝐶)
97, 8syl6eqr 2672 . . . . . . 7 (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
10 simpr 477 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
1110oveqd 6652 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑥𝑦) = (𝑥𝐻𝑦))
1211eleq2d 2685 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑓 ∈ (𝑥𝑦) ↔ 𝑓 ∈ (𝑥𝐻𝑦)))
1310oveqd 6652 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (𝑦𝑥) = (𝑦𝐻𝑥))
1413eleq2d 2685 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔 ∈ (𝑦𝑥) ↔ 𝑔 ∈ (𝑦𝐻𝑥)))
1512, 14anbi12d 746 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑓 ∈ (𝑥𝑦) ∧ 𝑔 ∈ (𝑦𝑥)) ↔ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥))))
16 simpl 473 . . . . . . . . . . . . 13 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
1716fveq2d 6182 . . . . . . . . . . . 12 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = (comp‘𝐶))
18 issect.o . . . . . . . . . . . 12 · = (comp‘𝐶)
1917, 18syl6eqr 2672 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (comp‘𝑐) = · )
2019oveqd 6652 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥) = (⟨𝑥, 𝑦· 𝑥))
2120oveqd 6652 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓))
2216fveq2d 6182 . . . . . . . . . . 11 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = (Id‘𝐶))
23 issect.i . . . . . . . . . . 11 1 = (Id‘𝐶)
2422, 23syl6eqr 2672 . . . . . . . . . 10 ((𝑐 = 𝐶 = 𝐻) → (Id‘𝑐) = 1 )
2524fveq1d 6180 . . . . . . . . 9 ((𝑐 = 𝐶 = 𝐻) → ((Id‘𝑐)‘𝑥) = ( 1𝑥))
2621, 25eqeq12d 2635 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → ((𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥) ↔ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥)))
2715, 26anbi12d 746 . . . . . . 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 3467 . . . . . 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 4707 . . . . 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 6702 . . . 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 16388 . . . 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))}))
32 fvex 6188 . . . . . 6 (Base‘𝐶) ∈ V
334, 32eqeltri 2695 . . . . 5 𝐵 ∈ V
3433, 33mpt2ex 7232 . . . 4 (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}) ∈ V
3530, 31, 34fvmpt 6269 . . 3 (𝐶 ∈ Cat → (Sect‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
362, 35syl 17 . 2 (𝜑 → (Sect‘𝐶) = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
371, 36syl5eq 2666 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 384   = wceq 1481  wcel 1988  Vcvv 3195  [wsbc 3429  cop 4174  {copab 4703  cfv 5876  (class class class)co 6635  cmpt2 6637  Basecbs 15838  Hom chom 15933  compcco 15934  Catccat 16306  Idccid 16307  Sectcsect 16385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-sect 16388
This theorem is referenced by:  sectfval  16392
  Copyright terms: Public domain W3C validator