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Theorem oppccatid 17609
Description: Lemma for oppccat 17612. (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 2733 . . . . 5 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
31, 2oppcbas 17607 . . . 4 (Baseβ€˜πΆ) = (Baseβ€˜π‘‚)
43a1i 11 . . 3 (𝐢 ∈ Cat β†’ (Baseβ€˜πΆ) = (Baseβ€˜π‘‚))
5 eqidd 2734 . . 3 (𝐢 ∈ Cat β†’ (Hom β€˜π‘‚) = (Hom β€˜π‘‚))
6 eqidd 2734 . . 3 (𝐢 ∈ Cat β†’ (compβ€˜π‘‚) = (compβ€˜π‘‚))
71fvexi 6860 . . . 4 𝑂 ∈ V
87a1i 11 . . 3 (𝐢 ∈ Cat β†’ 𝑂 ∈ V)
9 biid 261 . . 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 2733 . . . . 5 (Hom β€˜πΆ) = (Hom β€˜πΆ)
11 eqid 2733 . . . . 5 (Idβ€˜πΆ) = (Idβ€˜πΆ)
12 simpl 484 . . . . 5 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ 𝐢 ∈ Cat)
13 simpr 486 . . . . 5 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
142, 10, 11, 12, 13catidcl 17570 . . . 4 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ ((Idβ€˜πΆ)β€˜π‘¦) ∈ (𝑦(Hom β€˜πΆ)𝑦))
1510, 1oppchom 17604 . . . 4 (𝑦(Hom β€˜π‘‚)𝑦) = (𝑦(Hom β€˜πΆ)𝑦)
1614, 15eleqtrrdi 2845 . . 3 ((𝐢 ∈ Cat ∧ 𝑦 ∈ (Baseβ€˜πΆ)) β†’ ((Idβ€˜πΆ)β€˜π‘¦) ∈ (𝑦(Hom β€˜π‘‚)𝑦))
17 eqid 2733 . . . . 5 (compβ€˜πΆ) = (compβ€˜πΆ)
18 simpr1l 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†’ π‘₯ ∈ (Baseβ€˜πΆ))
19 simpr1r 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†’ 𝑦 ∈ (Baseβ€˜πΆ))
202, 17, 1, 18, 19, 19oppcco 17606 . . . 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 484 . . . . 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 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†’ 𝑓 ∈ (π‘₯(Hom β€˜π‘‚)𝑦))
2310, 1oppchom 17604 . . . . . 6 (π‘₯(Hom β€˜π‘‚)𝑦) = (𝑦(Hom β€˜πΆ)π‘₯)
2422, 23eleqtrdi 2844 . . . . 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 17572 . . . 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 2773 . . 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 1233 . . . . 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 17606 . . . 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 1265 . . . . . 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 17604 . . . . . 6 (𝑦(Hom β€˜π‘‚)𝑧) = (𝑧(Hom β€˜πΆ)𝑦)
3129, 30eleqtrdi 2844 . . . . 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 17571 . . . 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 2773 . . 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 17606 . . . . 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 17573 . . . . 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 2834 . . . 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 17604 . . . 4 (π‘₯(Hom β€˜π‘‚)𝑧) = (𝑧(Hom β€˜πΆ)π‘₯)
3836, 37eleqtrrdi 2845 . . 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 1234 . . . . . 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 1266 . . . . . . 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 17604 . . . . . . 7 (𝑧(Hom β€˜π‘‚)𝑀) = (𝑀(Hom β€˜πΆ)𝑧)
4240, 41eleqtrdi 2844 . . . . . 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 17574 . . . . 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 17606 . . . . 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 17606 . . . . 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 2784 . . . 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 17606 . . . . 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 7376 . . . 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 7377 . . . 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 2783 . . 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 17569 . 2 (𝐢 ∈ Cat β†’ (𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦))))
522, 11cidfn 17567 . . . . 5 (𝐢 ∈ Cat β†’ (Idβ€˜πΆ) Fn (Baseβ€˜πΆ))
53 dffn5 6905 . . . . 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 2744 . . 3 (𝐢 ∈ Cat β†’ ((Idβ€˜π‘‚) = (Idβ€˜πΆ) ↔ (Idβ€˜π‘‚) = (𝑦 ∈ (Baseβ€˜πΆ) ↦ ((Idβ€˜πΆ)β€˜π‘¦))))
5655anbi2d 630 . 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 257 1 (𝐢 ∈ Cat β†’ (𝑂 ∈ Cat ∧ (Idβ€˜π‘‚) = (Idβ€˜πΆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3447  βŸ¨cop 4596   ↦ cmpt 5192   Fn wfn 6495  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091  Hom chom 17152  compcco 17153  Catccat 17552  Idccid 17553  oppCatcoppc 17599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-tpos 8161  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-dec 12627  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-hom 17165  df-cco 17166  df-cat 17556  df-cid 17557  df-oppc 17600
This theorem is referenced by:  oppcid  17611  oppccat  17612
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