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Theorem invisoinvl 16431
Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 16409 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 2620 . . . 4 (comp‘𝐶) = (comp‘𝐶)
9 eqid 2620 . . . . . 6 (Id‘𝐶) = (Id‘𝐶)
101, 9, 3, 5idiso 16429 . . . . 5 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌(Iso‘𝐶)𝑌))
116a1i 11 . . . . . 6 (𝜑𝐼 = (Iso‘𝐶))
1211oveqd 6652 . . . . 5 (𝜑 → (𝑌𝐼𝑌) = (𝑌(Iso‘𝐶)𝑌))
1310, 12eleqtrrd 2702 . . . 4 (𝜑 → ((Id‘𝐶)‘𝑌) ∈ (𝑌𝐼𝑌))
141, 2, 3, 4, 5, 6, 7, 8, 5, 13invco 16412 . . 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 2620 . . . 4 (Hom ‘𝐶) = (Hom ‘𝐶)
161, 15, 6, 3, 4, 5isohom 16417 . . . . 5 (𝜑 → (𝑋𝐼𝑌) ⊆ (𝑋(Hom ‘𝐶)𝑌))
1716, 7sseldd 3596 . . . 4 (𝜑𝐹 ∈ (𝑋(Hom ‘𝐶)𝑌))
181, 15, 9, 3, 4, 8, 5, 17catlid 16325 . . 3 (𝜑 → (((Id‘𝐶)‘𝑌)(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑌)𝐹) = 𝐹)
192a1i 11 . . . . . . . 8 (𝜑𝑁 = (Inv‘𝐶))
2019oveqd 6652 . . . . . . 7 (𝜑 → (𝑌𝑁𝑌) = (𝑌(Inv‘𝐶)𝑌))
2120fveq1d 6180 . . . . . 6 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)))
221, 9, 3, 5idinv 16430 . . . . . 6 (𝜑 → ((𝑌(Inv‘𝐶)𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2321, 22eqtrd 2654 . . . . 5 (𝜑 → ((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌)) = ((Id‘𝐶)‘𝑌))
2423oveq2d 6651 . . . 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 16417 . . . . . 6 (𝜑 → (𝑌𝐼𝑋) ⊆ (𝑌(Hom ‘𝐶)𝑋))
261, 2, 3, 4, 5, 6invf 16409 . . . . . . 7 (𝜑 → (𝑋𝑁𝑌):(𝑋𝐼𝑌)⟶(𝑌𝐼𝑋))
2726, 7ffvelrnd 6346 . . . . . 6 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌𝐼𝑋))
2825, 27sseldd 3596 . . . . 5 (𝜑 → ((𝑋𝑁𝑌)‘𝐹) ∈ (𝑌(Hom ‘𝐶)𝑋))
291, 15, 9, 3, 5, 8, 4, 28catrid 16326 . . . 4 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((Id‘𝐶)‘𝑌)) = ((𝑋𝑁𝑌)‘𝐹))
3024, 29eqtrd 2654 . . 3 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(⟨𝑌, 𝑌⟩(comp‘𝐶)𝑋)((𝑌𝑁𝑌)‘((Id‘𝐶)‘𝑌))) = ((𝑋𝑁𝑌)‘𝐹))
3114, 18, 303brtr3d 4675 . 2 (𝜑𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹))
321, 2, 3, 5, 4invsym 16403 . 2 (𝜑 → (((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹𝐹(𝑋𝑁𝑌)((𝑋𝑁𝑌)‘𝐹)))
3331, 32mpbird 247 1 (𝜑 → ((𝑋𝑁𝑌)‘𝐹)(𝑌𝑁𝑋)𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cop 4174   class class class wbr 4644  cfv 5876  (class class class)co 6635  Basecbs 15838  Hom chom 15933  compcco 15934  Catccat 16306  Idccid 16307  Invcinv 16386  Isociso 16387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-cat 16310  df-cid 16311  df-sect 16388  df-inv 16389  df-iso 16390
This theorem is referenced by:  invisoinvr  16432  isocoinvid  16434
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