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Theorem oppccatid 17664
Description: Lemma for oppccat 17667. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypothesis
Ref Expression
oppcbas.1 𝑂 = (oppCatβ€˜πΆ)
Assertion
Ref Expression
oppccatid (𝐢 ∈ Cat β†’ (𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (Idβ€˜πΆ)))

Proof of Theorem oppccatid
Dummy variables 𝑓 𝑔 β„Ž 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oppcbas.1 . . . . 5 𝑂 = (oppCatβ€˜πΆ)
2 eqid 2732 . . . . 5 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
31, 2oppcbas 17662 . . . 4 (Baseβ€˜πΆ) = (Baseβ€˜π‘‚)
43a1i 11 . . 3 (𝐢 ∈ Cat β†’ (Baseβ€˜πΆ) = (Baseβ€˜π‘‚))
5 eqidd 2733 . . 3 (𝐢 ∈ Cat β†’ (Hom β€˜π‘‚) = (Hom β€˜π‘‚))
6 eqidd 2733 . . 3 (𝐢 ∈ Cat β†’ (compβ€˜π‘‚) = (compβ€˜π‘‚))
71fvexi 6905 . . . 4 𝑂 ∈ V
87a1i 11 . . 3 (𝐢 ∈ Cat β†’ 𝑂 ∈ V)
9 biid 260 . . 3 (((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀))) ↔ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀))))
10 eqid 2732 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
11 eqid 2732 . . . . 5 (Idβ€˜πΆ) = (Idβ€˜πΆ)
12 simpl 483 . . . . 5 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ 𝐢 ∈ Cat)
13 simpr 485 . . . . 5 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
142, 10, 11, 12, 13catidcl 17625 . . . 4 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ ((Idβ€˜πΆ)β€˜π‘¦) ∈ (𝑦(Hom β€˜πΆ)𝑦))
1510, 1oppchom 17659 . . . 4 (𝑦(Hom β€˜π‘‚)𝑦) = (𝑦(Hom β€˜πΆ)𝑦)
1614, 15eleqtrrdi 2844 . . 3 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ ((Idβ€˜πΆ)β€˜π‘¦) ∈ (𝑦(Hom β€˜π‘‚)𝑦))
17 eqid 2732 . . . . 5 (compβ€˜πΆ) = (compβ€˜πΆ)
18 simpr1l 1230 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ π‘₯ ∈ (Baseβ€˜πΆ))
19 simpr1r 1231 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
202, 17, 1, 18, 19, 19oppcco 17661 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (((Idβ€˜πΆ)β€˜π‘¦)(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑦)𝑓) = (𝑓(βŸ¨π‘¦, π‘¦βŸ©(compβ€˜πΆ)π‘₯)((Idβ€˜πΆ)β€˜π‘¦)))
21 simpl 483 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝐢 ∈ Cat)
22 simpr31 1263 . . . . . 6 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦))
2310, 1oppchom 17659 . . . . . 6 (π‘₯(Hom β€˜π‘‚)𝑦) = (𝑦(Hom β€˜πΆ)π‘₯)
2422, 23eleqtrdi 2843 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝑓 ∈ (𝑦(Hom β€˜πΆ)π‘₯))
252, 10, 11, 21, 19, 17, 18, 24catrid 17627 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (𝑓(βŸ¨π‘¦, π‘¦βŸ©(compβ€˜πΆ)π‘₯)((Idβ€˜πΆ)β€˜π‘¦)) = 𝑓)
2620, 25eqtrd 2772 . . 3 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (((Idβ€˜πΆ)β€˜π‘¦)(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑦)𝑓) = 𝑓)
27 simpr2l 1232 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝑧 ∈ (Baseβ€˜πΆ))
282, 17, 1, 19, 19, 27oppcco 17661 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (𝑔(βŸ¨π‘¦, π‘¦βŸ©(compβ€˜π‘‚)𝑧)((Idβ€˜πΆ)β€˜π‘¦)) = (((Idβ€˜πΆ)β€˜π‘¦)(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)𝑦)𝑔))
29 simpr32 1264 . . . . . 6 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧))
3010, 1oppchom 17659 . . . . . 6 (𝑦(Hom β€˜π‘‚)𝑧) = (𝑧(Hom β€˜πΆ)𝑦)
3129, 30eleqtrdi 2843 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝑔 ∈ (𝑧(Hom β€˜πΆ)𝑦))
322, 10, 11, 21, 27, 17, 19, 31catlid 17626 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (((Idβ€˜πΆ)β€˜π‘¦)(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)𝑦)𝑔) = 𝑔)
3328, 32eqtrd 2772 . . 3 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (𝑔(βŸ¨π‘¦, π‘¦βŸ©(compβ€˜π‘‚)𝑧)((Idβ€˜πΆ)β€˜π‘¦)) = 𝑔)
342, 17, 1, 18, 19, 27oppcco 17661 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑧)𝑓) = (𝑓(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)π‘₯)𝑔))
352, 10, 17, 21, 27, 19, 18, 31, 24catcocl 17628 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (𝑓(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)π‘₯)𝑔) ∈ (𝑧(Hom β€˜πΆ)π‘₯))
3634, 35eqeltrd 2833 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑧)𝑓) ∈ (𝑧(Hom β€˜πΆ)π‘₯))
3710, 1oppchom 17659 . . . 4 (π‘₯(Hom β€˜π‘‚)𝑧) = (𝑧(Hom β€˜πΆ)π‘₯)
3836, 37eleqtrrdi 2844 . . 3 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑧)𝑓) ∈ (π‘₯(Hom β€˜π‘‚)𝑧))
39 simpr2r 1233 . . . . . 6 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ 𝑀 ∈ (Baseβ€˜πΆ))
40 simpr33 1265 . . . . . . 7 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀))
4110, 1oppchom 17659 . . . . . . 7 (𝑧(Hom β€˜π‘‚)𝑀) = (𝑀(Hom β€˜πΆ)𝑧)
4240, 41eleqtrdi 2843 . . . . . 6 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ β„Ž ∈ (𝑀(Hom β€˜πΆ)𝑧))
432, 10, 17, 21, 39, 27, 19, 42, 31, 18, 24catass 17629 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ ((𝑓(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)π‘₯)𝑔)(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)π‘₯)β„Ž) = (𝑓(βŸ¨π‘€, π‘¦βŸ©(compβ€˜πΆ)π‘₯)(𝑔(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)𝑦)β„Ž)))
442, 17, 1, 18, 27, 39oppcco 17661 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (β„Ž(⟨π‘₯, π‘§βŸ©(compβ€˜π‘‚)𝑀)(𝑓(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)π‘₯)𝑔)) = ((𝑓(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)π‘₯)𝑔)(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)π‘₯)β„Ž))
452, 17, 1, 18, 19, 39oppcco 17661 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ ((𝑔(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)𝑦)β„Ž)(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑀)𝑓) = (𝑓(βŸ¨π‘€, π‘¦βŸ©(compβ€˜πΆ)π‘₯)(𝑔(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)𝑦)β„Ž)))
4643, 44, 453eqtr4rd 2783 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ ((𝑔(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)𝑦)β„Ž)(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑀)𝑓) = (β„Ž(⟨π‘₯, π‘§βŸ©(compβ€˜π‘‚)𝑀)(𝑓(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)π‘₯)𝑔)))
472, 17, 1, 19, 27, 39oppcco 17661 . . . . 5 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (β„Ž(βŸ¨π‘¦, π‘§βŸ©(compβ€˜π‘‚)𝑀)𝑔) = (𝑔(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)𝑦)β„Ž))
4847oveq1d 7423 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ ((β„Ž(βŸ¨π‘¦, π‘§βŸ©(compβ€˜π‘‚)𝑀)𝑔)(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑀)𝑓) = ((𝑔(βŸ¨π‘€, π‘§βŸ©(compβ€˜πΆ)𝑦)β„Ž)(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑀)𝑓))
4934oveq2d 7424 . . . 4 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ (β„Ž(⟨π‘₯, π‘§βŸ©(compβ€˜π‘‚)𝑀)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑧)𝑓)) = (β„Ž(⟨π‘₯, π‘§βŸ©(compβ€˜π‘‚)𝑀)(𝑓(βŸ¨π‘§, π‘¦βŸ©(compβ€˜πΆ)π‘₯)𝑔)))
5046, 48, 493eqtr4d 2782 . . 3 ((𝐢 ∈ Cat ∧ ((π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ)) ∧ (𝑧 ∈ (Baseβ€˜πΆ) ∧ 𝑀 ∈ (Baseβ€˜πΆ)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧) ∧ β„Ž ∈ (𝑧(Hom β€˜π‘‚)𝑀)))) β†’ ((β„Ž(βŸ¨π‘¦, π‘§βŸ©(compβ€˜π‘‚)𝑀)𝑔)(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑀)𝑓) = (β„Ž(⟨π‘₯, π‘§βŸ©(compβ€˜π‘‚)𝑀)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π‘‚)𝑧)𝑓)))
514, 5, 6, 8, 9, 16, 26, 33, 38, 50iscatd2 17624 . 2 (𝐢 ∈ Cat β†’ (𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦))))
522, 11cidfn 17622 . . . . 5 (𝐢 ∈ Cat β†’ (Idβ€˜πΆ) Fn (Baseβ€˜πΆ))
53 dffn5 6950 . . . . 5 ((Idβ€˜πΆ) Fn (Baseβ€˜πΆ) ↔ (Idβ€˜πΆ) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦)))
5452, 53sylib 217 . . . 4 (𝐢 ∈ Cat β†’ (Idβ€˜πΆ) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦)))
5554eqeq2d 2743 . . 3 (𝐢 ∈ Cat β†’ ((Idβ€˜π‘‚) = (Idβ€˜πΆ) ↔ (Idβ€˜π‘‚) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦))))
5655anbi2d 629 . 2 (𝐢 ∈ Cat β†’ ((𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (Idβ€˜πΆ)) ↔ (𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦)))))
5751, 56mpbird 256 1 (𝐢 ∈ Cat β†’ (𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (Idβ€˜πΆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474  βŸ¨cop 4634   ↦ cmpt 5231   Fn wfn 6538  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  Hom chom 17207  compcco 17208  Catccat 17607  Idccid 17608  oppCatcoppc 17654
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-hom 17220  df-cco 17221  df-cat 17611  df-cid 17612  df-oppc 17655
This theorem is referenced by:  oppcid  17666  oppccat  17667
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