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

Theorem issect 17001
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 16999 . . 3 (𝜑 → (𝑋𝑆𝑌) = {⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))})
109breqd 5050 . 2 (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺𝐹{⟨𝑓, 𝑔⟩ ∣ ((𝑓 ∈ (𝑋𝐻𝑌) ∧ 𝑔 ∈ (𝑌𝐻𝑋)) ∧ (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋))}𝐺))
11 oveq12 7139 . . . . . 6 ((𝑔 = 𝐺𝑓 = 𝐹) → (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹))
1211ancoms 462 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹))
1312eqeq1d 2823 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑔(⟨𝑋, 𝑌· 𝑋)𝑓) = ( 1𝑋) ↔ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)))
14 eqid 2821 . . . 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 5617 . . 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 1086 . . 3 ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)) ↔ ((𝐹 ∈ (𝑋𝐻𝑌) ∧ 𝐺 ∈ (𝑌𝐻𝑋)) ∧ (𝐺(⟨𝑋, 𝑌· 𝑋)𝐹) = ( 1𝑋)))
1715, 16bitr4i 281 . 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 290 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  cop 4546   class class class wbr 5039  {copab 5101  cfv 6328  (class class class)co 7130  Basecbs 16461  Hom chom 16554  compcco 16555  Catccat 16913  Idccid 16914  Sectcsect 16992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-sect 16995
This theorem is referenced by:  issect2  17002  sectcan  17003  sectco  17004  oppcsect  17026  sectmon  17030  monsect  17031  funcsect  17120  fucsect  17220  invfuc  17222  setcsect  17327  catciso  17345  rngcsect  44396  rngcsectALTV  44408  ringcsect  44447  ringcsectALTV  44471
  Copyright terms: Public domain W3C validator