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Theorem catccatid 18052
Description: Lemma for catccat 18054. (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 2733 . 2 (π‘ˆ ∈ 𝑉 β†’ (Hom β€˜πΆ) = (Hom β€˜πΆ))
4 eqidd 2733 . 2 (π‘ˆ ∈ 𝑉 β†’ (compβ€˜πΆ) = (compβ€˜πΆ))
5 catccatid.c . . . 4 𝐢 = (CatCatβ€˜π‘ˆ)
65fvexi 6902 . . 3 𝐢 ∈ V
76a1i 11 . 2 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ V)
8 biid 260 . 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 18047 . . . . . 6 (π‘ˆ ∈ 𝑉 β†’ 𝐡 = (π‘ˆ ∩ Cat))
11 inss2 4228 . . . . . 6 (π‘ˆ ∩ Cat) βŠ† Cat
1210, 11eqsstrdi 4035 . . . . 5 (π‘ˆ ∈ 𝑉 β†’ 𝐡 βŠ† Cat)
1312sselda 3981 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ Cat)
14 eqid 2732 . . . . 5 (idfuncβ€˜π‘₯) = (idfuncβ€˜π‘₯)
1514idfucl 17827 . . . 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 483 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ π‘ˆ ∈ 𝑉)
18 eqid 2732 . . . 4 (Hom β€˜πΆ) = (Hom β€˜πΆ)
19 simpr 485 . . . 4 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ π‘₯ ∈ 𝐡)
205, 1, 17, 18, 19, 19catchom 18049 . . 3 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯(Hom β€˜πΆ)π‘₯) = (π‘₯ Func π‘₯))
2116, 20eleqtrrd 2836 . 2 ((π‘ˆ ∈ 𝑉 ∧ π‘₯ ∈ 𝐡) β†’ (idfuncβ€˜π‘₯) ∈ (π‘₯(Hom β€˜πΆ)π‘₯))
22 simpl 483 . . . 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 2732 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
24 simpr1l 1230 . . . 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 1231 . . . 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 1263 . . . . 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 18049 . . . . 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 2835 . . . 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 591 . . . 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 18051 . . 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 17836 . . 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 2772 . 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 1232 . . . 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 1264 . . . . 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 18049 . . . . 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 2835 . . . 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 18051 . . 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 17837 . . 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 2772 . 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 17834 . . 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 18051 . . 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 18049 . . 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 2848 . 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 1265 . . . . . 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 1233 . . . . . . 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 18049 . . . . . 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 2835 . . . . 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 17835 . . . 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 17834 . . . . 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 18051 . . . 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 18051 . . . 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 2782 . . 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 18051 . . . 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 2782 . 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 17621 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   ∩ cin 3946  βŸ¨cop 4633   ↦ cmpt 5230  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Hom chom 17204  compcco 17205  Catccat 17604  Idccid 17605   Func cfunc 17800  idfunccidfu 17801   ∘func ccofu 17802  CatCatccatc 18044
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 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-ixp 8888  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-fz 13481  df-struct 17076  df-slot 17111  df-ndx 17123  df-base 17141  df-hom 17217  df-cco 17218  df-cat 17608  df-cid 17609  df-func 17804  df-idfu 17805  df-cofu 17806  df-catc 18045
This theorem is referenced by:  catcid  18053  catccat  18054
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