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Theorem fthinv 17818
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 2733 . . . 4 (Sectβ€˜πΆ) = (Sectβ€˜πΆ)
9 eqid 2733 . . . 4 (Sectβ€˜π·) = (Sectβ€˜π·)
101, 2, 3, 4, 5, 6, 7, 8, 9fthsect 17817 . . 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 17817 . . 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 632 . 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 17799 . . . . . . . 8 (𝐢 Faith 𝐷) βŠ† (𝐢 Func 𝐷)
1514ssbri 5151 . . . . . . 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 5107 . . . . . 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 17754 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝐢 Func 𝐷) β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
2018, 19syl 17 . . . 4 (πœ‘ β†’ (𝐢 ∈ Cat ∧ 𝐷 ∈ Cat))
2120simpld 496 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
221, 13, 21, 4, 5, 8isinv 17648 . 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 2733 . . 3 (Baseβ€˜π·) = (Baseβ€˜π·)
24 fthinv.t . . 3 𝐽 = (Invβ€˜π·)
2520simprd 497 . . 3 (πœ‘ β†’ 𝐷 ∈ Cat)
261, 23, 16funcf1 17757 . . . 4 (πœ‘ β†’ 𝐹:𝐡⟢(Baseβ€˜π·))
2726, 4ffvelcdmd 7037 . . 3 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ (Baseβ€˜π·))
2826, 5ffvelcdmd 7037 . . 3 (πœ‘ β†’ (πΉβ€˜π‘Œ) ∈ (Baseβ€˜π·))
2923, 24, 25, 27, 28, 9isinv 17648 . 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 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4593   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Hom chom 17149  Catccat 17549  Sectcsect 17632  Invcinv 17633   Func cfunc 17745   Faith cfth 17795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-ixp 8839  df-cat 17553  df-cid 17554  df-sect 17635  df-inv 17636  df-func 17749  df-fth 17797
This theorem is referenced by:  ffthiso  17821
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