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Theorem issect 16353
Description: The property "𝐹 is a section of 𝐺". (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
issect (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋))))

Proof of Theorem issect
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 issect.b . . . 4 𝐵 = (Base‘𝐶)
2 issect.h . . . 4 𝐻 = (Hom ‘𝐶)
3 issect.o . . . 4 · = (comp‘𝐶)
4 issect.i . . . 4 1 = (Id‘𝐶)
5 issect.s . . . 4 𝑆 = (Sect‘𝐶)
6 issect.c . . . 4 (𝜑𝐶 ∈ Cat)
7 issect.x . . . 4 (𝜑𝑋𝐵)
8 issect.y . . . 4 (𝜑𝑌𝐵)
91, 2, 3, 4, 5, 6, 7, 8sectfval 16351 . . 3 (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
109breqd 4634 . 2 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))}𝐺))
11 oveq12 6624 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝐹) → (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹))
1211ancoms 469 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹))
1312eqeq1d 2623 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋) ↔ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)))
14 eqid 2621 . . . 4 {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))} = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))}
1513, 14brab2a 5139 . . 3 (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))}𝐺 ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)))
16 df-3an 1038 . . 3 ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)))
1715, 16bitr4i 267 . 2 (𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))}𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)))
1810, 17syl6bb 276 1 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cop 4161   class class class wbr 4623  {copab 4682  cfv 5857  (class class class)co 6615  Basecbs 15800  Hom chom 15892  compcco 15893  Catccat 16265  Idccid 16266  Sectcsect 16344
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 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-sect 16347
This theorem is referenced by:  issect2  16354  sectcan  16355  sectco  16356  oppcsect  16378  sectmon  16382  monsect  16383  funcsect  16472  fucsect  16572  invfuc  16574  setcsect  16679  catciso  16697  rngcsect  41298  rngcsectALTV  41310  ringcsect  41349  ringcsectALTV  41373
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