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Theorem catccatid 18068
Description: Lemma for catccat 18070. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
catccatid.c 𝐢 = (CatCatβ€˜π‘ˆ)
catccatid.b 𝐡 = (Baseβ€˜πΆ)
Assertion
Ref Expression
catccatid (π‘ˆ ∈ 𝑉 β†’ (𝐢 ∈ Cat ∧ (Idβ€˜πΆ) = (π‘₯ ∈ 𝐡 ↦ (idfuncβ€˜π‘₯))))
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐢   π‘₯,π‘ˆ   π‘₯,𝑉
Proof of Theorem catccatid
Dummy variables 𝑓 𝑔 β„Ž 𝑀 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catccatid.b . . 3 𝐡 = (Baseβ€˜πΆ)
21a1i 11 . 2 (π‘ˆ ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜πΆ))
3 eqidd 2727 . 2 (π‘ˆ ∈ 𝑉 β†’ (Hom β€˜πΆ) = (Hom β€˜πΆ))
4 eqidd 2727 . 2 (π‘ˆ ∈ 𝑉 β†’ (compβ€˜πΆ) = (compβ€˜πΆ))
5 catccatid.c . . . 4 𝐢 = (CatCatβ€˜π‘ˆ)
65fvexi 6899 . . 3 𝐢 ∈ V
76a1i 11 . 2 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ V)
8 biid 261 . 2 (((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧))) ↔ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧))))
9 id 22 . . . . . . 7 (π‘ˆ ∈ 𝑉 β†’ π‘ˆ ∈ 𝑉)
105, 1, 9catcbas 18063 . . . . . 6 (π‘ˆ ∈ 𝑉 β†’ 𝐡 = (π‘ˆ ∩ Cat))
11 inss2 4224 . . . . . 6 (π‘ˆ ∩ Cat) βŠ† Cat
1210, 11eqsstrdi 4031 . . . . 5 (π‘ˆ ∈ 𝑉 β†’ 𝐡 βŠ† Cat)
1312sselda 3977 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ Cat)
14 eqid 2726 . . . . 5 (idfuncβ€˜π‘₯) = (idfuncβ€˜π‘₯)
1514idfucl 17840 . . . 4 (π‘₯ ∈ Cat β†’ (idfuncβ€˜π‘₯) ∈ (π‘₯ Func π‘₯))
1613, 15syl 17 . . 3 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ (idfuncβ€˜π‘₯) ∈ (π‘₯ Func π‘₯))
17 simpl 482 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ π‘ˆ ∈ 𝑉)
18 eqid 2726 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
19 simpr 484 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ 𝐡)
205, 1, 17, 18, 19, 19catchom 18065 . . 3 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯(Hom β€˜πΆ)π‘₯) = (π‘₯ Func π‘₯))
2116, 20eleqtrrd 2830 . 2 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ (idfuncβ€˜π‘₯) ∈ (π‘₯(Hom β€˜πΆ)π‘₯))
22 simpl 482 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ π‘ˆ ∈ 𝑉)
23 eqid 2726 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
24 simpr1l 1227 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ 𝑀 ∈ 𝐡)
25 simpr1r 1228 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ π‘₯ ∈ 𝐡)
26 simpr31 1260 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ 𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯))
275, 1, 22, 18, 24, 25catchom 18065 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑀(Hom β€˜πΆ)π‘₯) = (𝑀 Func π‘₯))
2826, 27eleqtrd 2829 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ 𝑓 ∈ (𝑀 Func π‘₯))
2925, 16syldan 590 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (idfuncβ€˜π‘₯) ∈ (π‘₯ Func π‘₯))
305, 1, 22, 23, 24, 25, 25, 28, 29catcco 18067 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((idfuncβ€˜π‘₯)(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)π‘₯)𝑓) = ((idfuncβ€˜π‘₯) ∘func 𝑓))
3128, 14cofulid 17849 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((idfuncβ€˜π‘₯) ∘func 𝑓) = 𝑓)
3230, 31eqtrd 2766 . 2 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((idfuncβ€˜π‘₯)(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)π‘₯)𝑓) = 𝑓)
33 simpr2l 1229 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ 𝑦 ∈ 𝐡)
34 simpr32 1261 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦))
355, 1, 22, 18, 25, 33catchom 18065 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (π‘₯(Hom β€˜πΆ)𝑦) = (π‘₯ Func 𝑦))
3634, 35eleqtrd 2829 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ 𝑔 ∈ (π‘₯ Func 𝑦))
375, 1, 22, 23, 25, 25, 33, 29, 36catcco 18067 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑔(⟨π‘₯, π‘₯⟩(compβ€˜πΆ)𝑦)(idfuncβ€˜π‘₯)) = (𝑔 ∘func (idfuncβ€˜π‘₯)))
3836, 14cofurid 17850 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑔 ∘func (idfuncβ€˜π‘₯)) = 𝑔)
3937, 38eqtrd 2766 . 2 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑔(⟨π‘₯, π‘₯⟩(compβ€˜πΆ)𝑦)(idfuncβ€˜π‘₯)) = 𝑔)
4028, 36cofucl 17847 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑔 ∘func 𝑓) ∈ (𝑀 Func 𝑦))
415, 1, 22, 23, 24, 25, 33, 28, 36catcco 18067 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑔(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑦)𝑓) = (𝑔 ∘func 𝑓))
425, 1, 22, 18, 24, 33catchom 18065 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑀(Hom β€˜πΆ)𝑦) = (𝑀 Func 𝑦))
4340, 41, 423eltr4d 2842 . 2 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑔(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑦)𝑓) ∈ (𝑀(Hom β€˜πΆ)𝑦))
44 simpr33 1262 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧))
45 simpr2r 1230 . . . . . . 7 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ 𝑧 ∈ 𝐡)
465, 1, 22, 18, 33, 45catchom 18065 . . . . . 6 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (𝑦(Hom β€˜πΆ)𝑧) = (𝑦 Func 𝑧))
4744, 46eleqtrd 2829 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ β„Ž ∈ (𝑦 Func 𝑧))
4828, 36, 47cofuass 17848 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((β„Ž ∘func 𝑔) ∘func 𝑓) = (β„Ž ∘func (𝑔 ∘func 𝑓)))
4936, 47cofucl 17847 . . . . 5 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (β„Ž ∘func 𝑔) ∈ (π‘₯ Func 𝑧))
505, 1, 22, 23, 24, 25, 45, 28, 49catcco 18067 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((β„Ž ∘func 𝑔)(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑧)𝑓) = ((β„Ž ∘func 𝑔) ∘func 𝑓))
515, 1, 22, 23, 24, 33, 45, 40, 47catcco 18067 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (β„Ž(βŸ¨π‘€, π‘¦βŸ©(compβ€˜πΆ)𝑧)(𝑔 ∘func 𝑓)) = (β„Ž ∘func (𝑔 ∘func 𝑓)))
5248, 50, 513eqtr4d 2776 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((β„Ž ∘func 𝑔)(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑧)𝑓) = (β„Ž(βŸ¨π‘€, π‘¦βŸ©(compβ€˜πΆ)𝑧)(𝑔 ∘func 𝑓)))
535, 1, 22, 23, 25, 33, 45, 36, 47catcco 18067 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (β„Ž(⟨π‘₯, π‘¦βŸ©(compβ€˜πΆ)𝑧)𝑔) = (β„Ž ∘func 𝑔))
5453oveq1d 7420 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((β„Ž(⟨π‘₯, π‘¦βŸ©(compβ€˜πΆ)𝑧)𝑔)(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑧)𝑓) = ((β„Ž ∘func 𝑔)(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑧)𝑓))
5541oveq2d 7421 . . 3 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ (β„Ž(βŸ¨π‘€, π‘¦βŸ©(compβ€˜πΆ)𝑧)(𝑔(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑦)𝑓)) = (β„Ž(βŸ¨π‘€, π‘¦βŸ©(compβ€˜πΆ)𝑧)(𝑔 ∘func 𝑓)))
5652, 54, 553eqtr4d 2776 . 2 ((π‘ˆ ∈ 𝑉 ∧ ((𝑀 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) ∧ (𝑦 ∈ 𝐡 ∧ 𝑧 ∈ 𝐡) ∧ (𝑓 ∈ (𝑀(Hom β€˜πΆ)π‘₯) ∧ 𝑔 ∈ (π‘₯(Hom β€˜πΆ)𝑦) ∧ β„Ž ∈ (𝑦(Hom β€˜πΆ)𝑧)))) β†’ ((β„Ž(⟨π‘₯, π‘¦βŸ©(compβ€˜πΆ)𝑧)𝑔)(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑧)𝑓) = (β„Ž(βŸ¨π‘€, π‘¦βŸ©(compβ€˜πΆ)𝑧)(𝑔(βŸ¨π‘€, π‘₯⟩(compβ€˜πΆ)𝑦)𝑓)))
572, 3, 4, 7, 8, 21, 32, 39, 43, 56iscatd2 17634 1 (π‘ˆ ∈ 𝑉 β†’ (𝐢 ∈ Cat ∧ (Idβ€˜πΆ) = (π‘₯ ∈ 𝐡 ↦ (idfuncβ€˜π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∩ cin 3942  βŸ¨cop 4629   ↦ cmpt 5224  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  Hom chom 17217  compcco 17218  Catccat 17617  Idccid 17618   Func cfunc 17813  idfunccidfu 17814   ∘func ccofu 17815  CatCatccatc 18060
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  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  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-nel 3041  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-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  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-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  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-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-fz 13491  df-struct 17089  df-slot 17124  df-ndx 17136  df-base 17154  df-hom 17230  df-cco 17231  df-cat 17621  df-cid 17622  df-func 17817  df-idfu 17818  df-cofu 17819  df-catc 18061
This theorem is referenced by:  catcid  18069  catccat  18070
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