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Theorem sectfval 16458
Description: Value of the section relation. (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
sectfval (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
Distinct variable groups:   𝑓,𝑔, 1   𝐶,𝑓,𝑔   𝜑,𝑓,𝑔   𝑓,𝐻,𝑔   · ,𝑓,𝑔   𝑓,𝑋,𝑔   𝑓,𝑌,𝑔
Allowed substitution hints:   𝐵(𝑓,𝑔)   𝑆(𝑓,𝑔)

Proof of Theorem sectfval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issect.b . . 3 𝐵 = (Base‘𝐶)
2 issect.h . . 3 𝐻 = (Hom ‘𝐶)
3 issect.o . . 3 · = (comp‘𝐶)
4 issect.i . . 3 1 = (Id‘𝐶)
5 issect.s . . 3 𝑆 = (Sect‘𝐶)
6 issect.c . . 3 (𝜑𝐶 ∈ Cat)
7 issect.x . . 3 (𝜑𝑋𝐵)
81, 2, 3, 4, 5, 6, 7, 7sectffval 16457 . 2 (𝜑𝑆 = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
9 simprl 809 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
10 simprr 811 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
119, 10oveq12d 6708 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1211eleq2d 2716 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑋𝐻𝑌)))
1310, 9oveq12d 6708 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝐻𝑥) = (𝑌𝐻𝑋))
1413eleq2d 2716 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑦𝐻𝑥) ↔ 𝑔 ∈ (𝑌𝐻𝑋)))
1512, 14anbi12d 747 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
169, 10opeq12d 4441 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
1716, 9oveq12d 6708 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (⟨𝑥, 𝑦· 𝑥) = (⟨𝑋, 𝑌· 𝑋))
1817oveqd 6707 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓))
199fveq2d 6233 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( 1𝑥) = ( 1𝑋))
2018, 19eqeq12d 2666 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥) ↔ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋)))
2115, 20anbi12d 747 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥)) ↔ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))))
2221opabbidv 4749 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
23 issect.y . 2 (𝜑𝑌𝐵)
24 ovex 6718 . . . . 5 (𝑋𝐻𝑌) ∈ V
25 ovex 6718 . . . . 5 (𝑌𝐻𝑋) ∈ V
2624, 25xpex 7004 . . . 4 ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ∈ V
27 opabssxp 5227 . . . 4 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))
2826, 27ssexi 4836 . . 3 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))} ∈ V
2928a1i 11 . 2 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))} ∈ V)
308, 22, 7, 23, 29ovmpt2d 6830 1 (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216  {copab 4745   × cxp 5141  cfv 5926  (class class class)co 6690  Basecbs 15904  Hom chom 15999  compcco 16000  Catccat 16372  Idccid 16373  Sectcsect 16451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-sect 16454
This theorem is referenced by:  sectss  16459  issect  16460  dfiso2  16479
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