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

Theorem fthinv 17886
Description: A faithful functor reflects inverses. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
fthsect.b 𝐡 = (Baseβ€˜πΆ)
fthsect.h 𝐻 = (Hom β€˜πΆ)
fthsect.f (πœ‘ β†’ 𝐹(𝐢 Faith 𝐷)𝐺)
fthsect.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
fthsect.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
fthsect.m (πœ‘ β†’ 𝑀 ∈ (π‘‹π»π‘Œ))
fthsect.n (πœ‘ β†’ 𝑁 ∈ (π‘Œπ»π‘‹))
fthinv.s 𝐼 = (Invβ€˜πΆ)
fthinv.t 𝐽 = (Invβ€˜π·)
Assertion
Ref Expression
fthinv (πœ‘ β†’ (𝑀(π‘‹πΌπ‘Œ)𝑁 ↔ ((π‘‹πΊπ‘Œ)β€˜π‘€)((πΉβ€˜π‘‹)𝐽(πΉβ€˜π‘Œ))((π‘ŒπΊπ‘‹)β€˜π‘)))

Proof of Theorem fthinv
StepHypRef Expression
1 fthsect.b . . . 4 𝐡 = (Baseβ€˜πΆ)
2 fthsect.h . . . 4 𝐻 = (Hom β€˜πΆ)
3 fthsect.f . . . 4 (πœ‘ β†’ 𝐹(𝐢 Faith 𝐷)𝐺)
4 fthsect.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐡)
5 fthsect.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝐡)
6 fthsect.m . . . 4 (πœ‘ β†’ 𝑀 ∈ (π‘‹π»π‘Œ))
7 fthsect.n . . . 4 (πœ‘ β†’ 𝑁 ∈ (π‘Œπ»π‘‹))
8 eqid 2731 . . . 4 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
9 eqid 2731 . . . 4 (Sectβ€˜π·) = (Sectβ€˜π·)
101, 2, 3, 4, 5, 6, 7, 8, 9fthsect 17885 . . 3 (πœ‘ β†’ (𝑀(𝑋(Sectβ€˜πΆ)π‘Œ)𝑁 ↔ ((π‘‹πΊπ‘Œ)β€˜π‘€)((πΉβ€˜π‘‹)(Sectβ€˜π·)(πΉβ€˜π‘Œ))((π‘ŒπΊπ‘‹)β€˜π‘)))
111, 2, 3, 5, 4, 7, 6, 8, 9fthsect 17885 . . 3 (πœ‘ β†’ (𝑁(π‘Œ(Sectβ€˜πΆ)𝑋)𝑀 ↔ ((π‘ŒπΊπ‘‹)β€˜π‘)((πΉβ€˜π‘Œ)(Sectβ€˜π·)(πΉβ€˜π‘‹))((π‘‹πΊπ‘Œ)β€˜π‘€)))
1210, 11anbi12d 630 . 2 (πœ‘ β†’ ((𝑀(𝑋(Sectβ€˜πΆ)π‘Œ)𝑁 ∧ 𝑁(π‘Œ(Sectβ€˜πΆ)𝑋)𝑀) ↔ (((π‘‹πΊπ‘Œ)β€˜π‘€)((πΉβ€˜π‘‹)(Sectβ€˜π·)(πΉβ€˜π‘Œ))((π‘ŒπΊπ‘‹)β€˜π‘) ∧ ((π‘ŒπΊπ‘‹)β€˜π‘)((πΉβ€˜π‘Œ)(Sectβ€˜π·)(πΉβ€˜π‘‹))((π‘‹πΊπ‘Œ)β€˜π‘€))))
13 fthinv.s . . 3 𝐼 = (Invβ€˜πΆ)
14 fthfunc 17867 . . . . . . . 8 (𝐢 Faith 𝐷) βŠ† (𝐢 Func 𝐷)
1514ssbri 5193 . . . . . . 7 (𝐹(𝐢 Faith 𝐷)𝐺 β†’ 𝐹(𝐢 Func 𝐷)𝐺)
163, 15syl 17 . . . . . 6 (πœ‘ β†’ 𝐹(𝐢 Func 𝐷)𝐺)
17 df-br 5149 . . . . . 6 (𝐹(𝐢 Func 𝐷)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷))
1816, 17sylib 217 . . . . 5 (πœ‘ β†’ ⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷))
19 funcrcl 17820 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
2018, 19syl 17 . . . 4 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
2120simpld 494 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
221, 13, 21, 4, 5, 8isinv 17714 . 2 (πœ‘ β†’ (𝑀(π‘‹πΌπ‘Œ)𝑁 ↔ (𝑀(𝑋(Sectβ€˜πΆ)π‘Œ)𝑁 ∧ 𝑁(π‘Œ(Sectβ€˜πΆ)𝑋)𝑀)))
23 eqid 2731 . . 3 (Baseβ€˜π·) = (Baseβ€˜π·)
24 fthinv.t . . 3 𝐽 = (Invβ€˜π·)
2520simprd 495 . . 3 (πœ‘ β†’ 𝐷 ∈ Cat)
261, 23, 16funcf1 17823 . . . 4 (πœ‘ β†’ 𝐹:𝐡⟢(Baseβ€˜π·))
2726, 4ffvelcdmd 7087 . . 3 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ (Baseβ€˜π·))
2826, 5ffvelcdmd 7087 . . 3 (πœ‘ β†’ (πΉβ€˜π‘Œ) ∈ (Baseβ€˜π·))
2923, 24, 25, 27, 28, 9isinv 17714 . 2 (πœ‘ β†’ (((π‘‹πΊπ‘Œ)β€˜π‘€)((πΉβ€˜π‘‹)𝐽(πΉβ€˜π‘Œ))((π‘ŒπΊπ‘‹)β€˜π‘) ↔ (((π‘‹πΊπ‘Œ)β€˜π‘€)((πΉβ€˜π‘‹)(Sectβ€˜π·)(πΉβ€˜π‘Œ))((π‘ŒπΊπ‘‹)β€˜π‘) ∧ ((π‘ŒπΊπ‘‹)β€˜π‘)((πΉβ€˜π‘Œ)(Sectβ€˜π·)(πΉβ€˜π‘‹))((π‘‹πΊπ‘Œ)β€˜π‘€))))
3012, 22, 293bitr4d 311 1 (πœ‘ β†’ (𝑀(π‘‹πΌπ‘Œ)𝑁 ↔ ((π‘‹πΊπ‘Œ)β€˜π‘€)((πΉβ€˜π‘‹)𝐽(πΉβ€˜π‘Œ))((π‘ŒπΊπ‘‹)β€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βŸ¨cop 4634   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  Basecbs 17151  Hom chom 17215  Catccat 17615  Sectcsect 17698  Invcinv 17699   Func cfunc 17811   Faith cfth 17863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8828  df-ixp 8898  df-cat 17619  df-cid 17620  df-sect 17701  df-inv 17702  df-func 17815  df-fth 17865
This theorem is referenced by:  ffthiso  17889
  Copyright terms: Public domain W3C validator