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Theorem invisoinvl 17770
Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17748 and is therefore denoted by ((π‘‹π‘π‘Œ)β€˜πΉ), 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 𝐡 = (Baseβ€˜πΆ)
invisoinv.i 𝐼 = (Isoβ€˜πΆ)
invisoinv.n 𝑁 = (Invβ€˜πΆ)
invisoinv.c (πœ‘ β†’ 𝐢 ∈ Cat)
invisoinv.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
invisoinv.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
invisoinv.f (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
Assertion
Ref Expression
invisoinvl (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹)

Proof of Theorem invisoinvl
StepHypRef Expression
1 invisoinv.b . . . 4 𝐡 = (Baseβ€˜πΆ)
2 invisoinv.n . . . 4 𝑁 = (Invβ€˜πΆ)
3 invisoinv.c . . . 4 (πœ‘ β†’ 𝐢 ∈ Cat)
4 invisoinv.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐡)
5 invisoinv.y . . . 4 (πœ‘ β†’ π‘Œ ∈ 𝐡)
6 invisoinv.i . . . 4 𝐼 = (Isoβ€˜πΆ)
7 invisoinv.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (π‘‹πΌπ‘Œ))
8 eqid 2725 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
9 eqid 2725 . . . . . 6 (Idβ€˜πΆ) = (Idβ€˜πΆ)
101, 9, 3, 5idiso 17768 . . . . 5 (πœ‘ β†’ ((Idβ€˜πΆ)β€˜π‘Œ) ∈ (π‘Œ(Isoβ€˜πΆ)π‘Œ))
116a1i 11 . . . . . 6 (πœ‘ β†’ 𝐼 = (Isoβ€˜πΆ))
1211oveqd 7432 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘Œ) = (π‘Œ(Isoβ€˜πΆ)π‘Œ))
1310, 12eleqtrrd 2828 . . . 4 (πœ‘ β†’ ((Idβ€˜πΆ)β€˜π‘Œ) ∈ (π‘ŒπΌπ‘Œ))
141, 2, 3, 4, 5, 6, 7, 8, 5, 13invco 17751 . . 3 (πœ‘ β†’ (((Idβ€˜πΆ)β€˜π‘Œ)(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)π‘Œ)𝐹)(π‘‹π‘π‘Œ)(((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘Œ, π‘ŒβŸ©(compβ€˜πΆ)𝑋)((π‘Œπ‘π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ))))
15 eqid 2725 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
161, 15, 6, 3, 4, 5isohom 17756 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
1716, 7sseldd 3973 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
181, 15, 9, 3, 4, 8, 5, 17catlid 17660 . . 3 (πœ‘ β†’ (((Idβ€˜πΆ)β€˜π‘Œ)(βŸ¨π‘‹, π‘ŒβŸ©(compβ€˜πΆ)π‘Œ)𝐹) = 𝐹)
192a1i 11 . . . . . . . 8 (πœ‘ β†’ 𝑁 = (Invβ€˜πΆ))
2019oveqd 7432 . . . . . . 7 (πœ‘ β†’ (π‘Œπ‘π‘Œ) = (π‘Œ(Invβ€˜πΆ)π‘Œ))
2120fveq1d 6893 . . . . . 6 (πœ‘ β†’ ((π‘Œπ‘π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ)) = ((π‘Œ(Invβ€˜πΆ)π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ)))
221, 9, 3, 5idinv 17769 . . . . . 6 (πœ‘ β†’ ((π‘Œ(Invβ€˜πΆ)π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ)) = ((Idβ€˜πΆ)β€˜π‘Œ))
2321, 22eqtrd 2765 . . . . 5 (πœ‘ β†’ ((π‘Œπ‘π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ)) = ((Idβ€˜πΆ)β€˜π‘Œ))
2423oveq2d 7431 . . . 4 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘Œ, π‘ŒβŸ©(compβ€˜πΆ)𝑋)((π‘Œπ‘π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ))) = (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘Œ, π‘ŒβŸ©(compβ€˜πΆ)𝑋)((Idβ€˜πΆ)β€˜π‘Œ)))
251, 15, 6, 3, 5, 4isohom 17756 . . . . . 6 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
261, 2, 3, 4, 5, 6invf 17748 . . . . . . 7 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
2726, 7ffvelcdmd 7089 . . . . . 6 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2825, 27sseldd 3973 . . . . 5 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘Œ(Hom β€˜πΆ)𝑋))
291, 15, 9, 3, 5, 8, 4, 28catrid 17661 . . . 4 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘Œ, π‘ŒβŸ©(compβ€˜πΆ)𝑋)((Idβ€˜πΆ)β€˜π‘Œ)) = ((π‘‹π‘π‘Œ)β€˜πΉ))
3024, 29eqtrd 2765 . . 3 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(βŸ¨π‘Œ, π‘ŒβŸ©(compβ€˜πΆ)𝑋)((π‘Œπ‘π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ))) = ((π‘‹π‘π‘Œ)β€˜πΉ))
3114, 18, 303brtr3d 5174 . 2 (πœ‘ β†’ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
321, 2, 3, 5, 4invsym 17742 . 2 (πœ‘ β†’ (((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹 ↔ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ)))
3331, 32mpbird 256 1 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ)(π‘Œπ‘π‘‹)𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4630   class class class wbr 5143  β€˜cfv 6542  (class class class)co 7415  Basecbs 17177  Hom chom 17241  compcco 17242  Catccat 17641  Idccid 17642  Invcinv 17725  Isociso 17726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-cat 17645  df-cid 17646  df-sect 17727  df-inv 17728  df-iso 17729
This theorem is referenced by:  invisoinvr  17771  isocoinvid  17773
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