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Theorem oppccatid 17670
Description: Lemma for oppccat 17673. (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 2731 . . . . 5 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
31, 2oppcbas 17668 . . . 4 (Baseβ€˜πΆ) = (Baseβ€˜π‘‚)
43a1i 11 . . 3 (𝐢 ∈ Cat β†’ (Baseβ€˜πΆ) = (Baseβ€˜π‘‚))
5 eqidd 2732 . . 3 (𝐢 ∈ Cat β†’ (Hom β€˜π‘‚) = (Hom β€˜π‘‚))
6 eqidd 2732 . . 3 (𝐢 ∈ Cat β†’ (compβ€˜π‘‚) = (compβ€˜π‘‚))
71fvexi 6906 . . . 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 2731 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
11 eqid 2731 . . . . 5 (Idβ€˜πΆ) = (Idβ€˜πΆ)
12 simpl 482 . . . . 5 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ 𝐢 ∈ Cat)
13 simpr 484 . . . . 5 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
142, 10, 11, 12, 13catidcl 17631 . . . 4 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ ((Idβ€˜πΆ)β€˜π‘¦) ∈ (𝑦(Hom β€˜πΆ)𝑦))
1510, 1oppchom 17665 . . . 4 (𝑦(Hom β€˜π‘‚)𝑦) = (𝑦(Hom β€˜πΆ)𝑦)
1614, 15eleqtrrdi 2843 . . 3 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ ((Idβ€˜πΆ)β€˜π‘¦) ∈ (𝑦(Hom β€˜π‘‚)𝑦))
17 eqid 2731 . . . . 5 (compβ€˜πΆ) = (compβ€˜πΆ)
18 simpr1l 1229 . . . . 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 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†’ 𝑦 ∈ (Baseβ€˜πΆ))
202, 17, 1, 18, 19, 19oppcco 17667 . . . 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 482 . . . . 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 1262 . . . . . 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 17665 . . . . . 6 (π‘₯(Hom β€˜π‘‚)𝑦) = (𝑦(Hom β€˜πΆ)π‘₯)
2422, 23eleqtrdi 2842 . . . . 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 17633 . . . 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 2771 . . 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 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†’ 𝑧 ∈ (Baseβ€˜πΆ))
282, 17, 1, 19, 19, 27oppcco 17667 . . . 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 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†’ 𝑔 ∈ (𝑦(Hom β€˜π‘‚)𝑧))
3010, 1oppchom 17665 . . . . . 6 (𝑦(Hom β€˜π‘‚)𝑧) = (𝑧(Hom β€˜πΆ)𝑦)
3129, 30eleqtrdi 2842 . . . . 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 17632 . . . 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 2771 . . 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 17667 . . . . 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 17634 . . . . 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 2832 . . . 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 17665 . . . 4 (π‘₯(Hom β€˜π‘‚)𝑧) = (𝑧(Hom β€˜πΆ)π‘₯)
3836, 37eleqtrrdi 2843 . . 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 1232 . . . . . 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 1264 . . . . . . 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 17665 . . . . . . 7 (𝑧(Hom β€˜π‘‚)𝑀) = (𝑀(Hom β€˜πΆ)𝑧)
4240, 41eleqtrdi 2842 . . . . . 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 17635 . . . . 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 17667 . . . . 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 17667 . . . . 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 2782 . . . 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 17667 . . . . 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 7427 . . . 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 7428 . . . 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 2781 . . 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 17630 . 2 (𝐢 ∈ Cat β†’ (𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦))))
522, 11cidfn 17628 . . . . 5 (𝐢 ∈ Cat β†’ (Idβ€˜πΆ) Fn (Baseβ€˜πΆ))
53 dffn5 6951 . . . . 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 2742 . . 3 (𝐢 ∈ Cat β†’ ((Idβ€˜π‘‚) = (Idβ€˜πΆ) ↔ (Idβ€˜π‘‚) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦))))
5655anbi2d 628 . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  Vcvv 3473  βŸ¨cop 4635   ↦ cmpt 5232   Fn wfn 6539  β€˜cfv 6544  (class class class)co 7412  Basecbs 17149  Hom chom 17213  compcco 17214  Catccat 17613  Idccid 17614  oppCatcoppc 17660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-tpos 8214  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-er 8706  df-en 8943  df-dom 8944  df-sdom 8945  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-hom 17226  df-cco 17227  df-cat 17617  df-cid 17618  df-oppc 17661
This theorem is referenced by:  oppcid  17672  oppccat  17673
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