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Theorem invisoinvl 16215
Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 16193 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-2017.)
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 2605 . . . 4 (comp‘𝐶) = (comp‘𝐶)
9 eqid 2605 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
101, 9, 3, 5idiso 16213 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌))
116a1i 11 . . . . . 6 (𝜑𝐼 = (Iso‘𝐶))
1211oveqd 6540 . . . . 5 (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌))
1310, 12eleqtrrd 2686 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌))
141, 2, 3, 4, 5, 6, 7, 8, 5, 13invco 16196 . . 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 2605 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
161, 15, 6, 3, 4, 5isohom 16201 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1716, 7sseldd 3564 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 15, 9, 3, 4, 8, 5, 17catlid 16109 . . 3 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
192a1i 11 . . . . . . . 8 (𝜑𝑁 = (Inv‘𝐶))
2019oveqd 6540 . . . . . . 7 (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌))
2120fveq1d 6086 . . . . . 6 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)))
221, 9, 3, 5idinv 16214 . . . . . 6 (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2321, 22eqtrd 2639 . . . . 5 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2423oveq2d 6539 . . . 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 16201 . . . . . 6 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
261, 2, 3, 4, 5, 6invf 16193 . . . . . . 7 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
2726, 7ffvelrnd 6249 . . . . . 6 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2825, 27sseldd 3564 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
291, 15, 9, 3, 5, 8, 4, 28catrid 16110 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹))
3024, 29eqtrd 2639 . . 3 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹))
3114, 18, 303brtr3d 4604 . 2 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
321, 2, 3, 5, 4invsym 16187 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
3331, 32mpbird 245 1 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1975  cop 4126   class class class wbr 4573  cfv 5786  (class class class)co 6523  Basecbs 15637  Hom chom 15721  compcco 15722  Catccat 16090  Idccid 16091  Invcinv 16170  Isociso 16171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-cat 16094  df-cid 16095  df-sect 16172  df-inv 16173  df-iso 16174
This theorem is referenced by:  invisoinvr  16216  isocoinvid  16218
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