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Theorem invisoinvl 17746
Description: The inverse of an isomorphism 𝐹 (which is unique because of invf 17724 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 2726 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
9 eqid 2726 . . . . . 6 (Idβ€˜πΆ) = (Idβ€˜πΆ)
101, 9, 3, 5idiso 17744 . . . . 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 7422 . . . . 5 (πœ‘ β†’ (π‘ŒπΌπ‘Œ) = (π‘Œ(Isoβ€˜πΆ)π‘Œ))
1310, 12eleqtrrd 2830 . . . 4 (πœ‘ β†’ ((Idβ€˜πΆ)β€˜π‘Œ) ∈ (π‘ŒπΌπ‘Œ))
141, 2, 3, 4, 5, 6, 7, 8, 5, 13invco 17727 . . 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 2726 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
161, 15, 6, 3, 4, 5isohom 17732 . . . . 5 (πœ‘ β†’ (π‘‹πΌπ‘Œ) βŠ† (𝑋(Hom β€˜πΆ)π‘Œ))
1716, 7sseldd 3978 . . . 4 (πœ‘ β†’ 𝐹 ∈ (𝑋(Hom β€˜πΆ)π‘Œ))
181, 15, 9, 3, 4, 8, 5, 17catlid 17636 . . 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 7422 . . . . . . 7 (πœ‘ β†’ (π‘Œπ‘π‘Œ) = (π‘Œ(Invβ€˜πΆ)π‘Œ))
2120fveq1d 6887 . . . . . 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 17745 . . . . . 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 2766 . . . . 5 (πœ‘ β†’ ((π‘Œπ‘π‘Œ)β€˜((Idβ€˜πΆ)β€˜π‘Œ)) = ((Idβ€˜πΆ)β€˜π‘Œ))
2423oveq2d 7421 . . . 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 17732 . . . . . 6 (πœ‘ β†’ (π‘ŒπΌπ‘‹) βŠ† (π‘Œ(Hom β€˜πΆ)𝑋))
261, 2, 3, 4, 5, 6invf 17724 . . . . . . 7 (πœ‘ β†’ (π‘‹π‘π‘Œ):(π‘‹πΌπ‘Œ)⟢(π‘ŒπΌπ‘‹))
2726, 7ffvelcdmd 7081 . . . . . 6 (πœ‘ β†’ ((π‘‹π‘π‘Œ)β€˜πΉ) ∈ (π‘ŒπΌπ‘‹))
2825, 27sseldd 3978 . . . . 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 17637 . . . 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 2766 . . 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 5172 . 2 (πœ‘ β†’ 𝐹(π‘‹π‘π‘Œ)((π‘‹π‘π‘Œ)β€˜πΉ))
321, 2, 3, 5, 4invsym 17718 . 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 257 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 4629   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  Hom chom 17217  compcco 17218  Catccat 17617  Idccid 17618  Invcinv 17701  Isociso 17702
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-cat 17621  df-cid 17622  df-sect 17703  df-inv 17704  df-iso 17705
This theorem is referenced by:  invisoinvr  17747  isocoinvid  17749
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