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Theorem invisoinvl 17051
Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17029 and is therefore denoted by ((𝑋𝑁𝑌)‘𝐹), see also remark 3.12 in [Adamek] p. 28) is invers to the isomorphism. (Contributed by AV, 9-Apr-2020.)
Hypotheses
Ref Expression
invisoinv.b 𝐵 = (Base‘𝐶)
invisoinv.i 𝐼 = (Iso‘𝐶)
invisoinv.n 𝑁 = (Inv‘𝐶)
invisoinv.c (𝜑𝐶 ∈ Cat)
invisoinv.x (𝜑𝑋𝐵)
invisoinv.y (𝜑𝑌𝐵)
invisoinv.f (𝜑𝐹 ∈ (𝑋𝐼𝑌))
Assertion
Ref Expression
invisoinvl (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)

Proof of Theorem invisoinvl
StepHypRef Expression
1 invisoinv.b . . . 4 𝐵 = (Base‘𝐶)
2 invisoinv.n . . . 4 𝑁 = (Inv‘𝐶)
3 invisoinv.c . . . 4 (𝜑𝐶 ∈ Cat)
4 invisoinv.x . . . 4 (𝜑𝑋𝐵)
5 invisoinv.y . . . 4 (𝜑𝑌𝐵)
6 invisoinv.i . . . 4 𝐼 = (Iso‘𝐶)
7 invisoinv.f . . . 4 (𝜑𝐹 ∈ (𝑋𝐼𝑌))
8 eqid 2822 . . . 4 (comp‘𝐶) = (comp‘𝐶)
9 eqid 2822 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
101, 9, 3, 5idiso 17049 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌))
116a1i 11 . . . . . 6 (𝜑𝐼 = (Iso‘𝐶))
1211oveqd 7157 . . . . 5 (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌))
1310, 12eleqtrrd 2917 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌))
141, 2, 3, 4, 5, 6, 7, 8, 5, 13invco 17032 . . 3 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹)(𝑋𝑁𝑌)(((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))))
15 eqid 2822 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
161, 15, 6, 3, 4, 5isohom 17037 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1716, 7sseldd 3943 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 15, 9, 3, 4, 8, 5, 17catlid 16945 . . 3 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
192a1i 11 . . . . . . . 8 (𝜑𝑁 = (Inv‘𝐶))
2019oveqd 7157 . . . . . . 7 (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌))
2120fveq1d 6654 . . . . . 6 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)))
221, 9, 3, 5idinv 17050 . . . . . 6 (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2321, 22eqtrd 2857 . . . . 5 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2423oveq2d 7156 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)))
251, 15, 6, 3, 5, 4isohom 17037 . . . . . 6 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
261, 2, 3, 4, 5, 6invf 17029 . . . . . . 7 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
2726, 7ffvelrnd 6834 . . . . . 6 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2825, 27sseldd 3943 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
291, 15, 9, 3, 5, 8, 4, 28catrid 16946 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹))
3024, 29eqtrd 2857 . . 3 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹))
3114, 18, 303brtr3d 5073 . 2 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
321, 2, 3, 5, 4invsym 17023 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
3331, 32mpbird 260 1 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  cop 4545   class class class wbr 5042  cfv 6334  (class class class)co 7140  Basecbs 16474  Hom chom 16567  compcco 16568  Catccat 16926  Idccid 16927  Invcinv 17006  Isociso 17007
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-cat 16930  df-cid 16931  df-sect 17008  df-inv 17009  df-iso 17010
This theorem is referenced by:  invisoinvr  17052  isocoinvid  17054
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