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Theorem sectfval 16332
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 16331 . 2 (𝜑𝑆 = (𝑥𝐵, 𝑦𝐵 ↦ {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ∧ (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥))}))
9 simprl 793 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑥 = 𝑋)
10 simprr 795 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑦 = 𝑌)
119, 10oveq12d 6622 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
1211eleq2d 2684 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑋𝐻𝑌)))
1310, 9oveq12d 6622 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑦𝐻𝑥) = (𝑌𝐻𝑋))
1413eleq2d 2684 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔 ∈ (𝑦𝐻𝑥) ↔ 𝑔 ∈ (𝑌𝐻𝑋)))
1512, 14anbi12d 746 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑥)) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋))))
169, 10opeq12d 4378 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑌⟩)
1716, 9oveq12d 6622 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (⟨𝑥, 𝑦· 𝑥) = (⟨𝑋, 𝑌· 𝑋))
1817oveqd 6621 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → (𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓))
199fveq2d 6152 . . . . 5 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ( 1𝑥) = ( 1𝑋))
2018, 19eqeq12d 2636 . . . 4 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → ((𝑔(⟨𝑥, 𝑦· 𝑥)𝑓) = ( 1𝑥) ↔ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋)))
2115, 20anbi12d 746 . . 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 4678 . 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 6632 . . . . 5 (𝑋𝐻𝑌) ∈ V
25 ovex 6632 . . . . 5 (𝑌𝐻𝑋) ∈ V
2624, 25xpex 6915 . . . 4 ((𝑋𝐻𝑌) × (𝑌𝐻𝑋)) ∈ V
27 opabssxp 5154 . . . 4 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))} ⊆ ((𝑋𝐻𝑌) × (𝑌𝐻𝑋))
2826, 27ssexi 4763 . . 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 6741 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 384   = wceq 1480  wcel 1987  Vcvv 3186  cop 4154  {copab 4672   × cxp 5072  cfv 5847  (class class class)co 6604  Basecbs 15781  Hom chom 15873  compcco 15874  Catccat 16246  Idccid 16247  Sectcsect 16325
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-sect 16328
This theorem is referenced by:  sectss  16333  issect  16334  dfiso2  16353
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