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Theorem resscatc 16968
Description: The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCat‘𝑈 categories for different 𝑈 are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resscatc.c 𝐶 = (CatCat‘𝑈)
resscatc.d 𝐷 = (CatCat‘𝑉)
resscatc.1 (𝜑𝑈𝑊)
resscatc.2 (𝜑𝑉𝑈)
Assertion
Ref Expression
resscatc (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))

Proof of Theorem resscatc
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resscatc.d . . . . . 6 𝐷 = (CatCat‘𝑉)
2 eqid 2769 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
3 resscatc.1 . . . . . . . 8 (𝜑𝑈𝑊)
4 resscatc.2 . . . . . . . 8 (𝜑𝑉𝑈)
53, 4ssexd 4935 . . . . . . 7 (𝜑𝑉 ∈ V)
65adantr 473 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑉 ∈ V)
7 eqid 2769 . . . . . 6 (Hom ‘𝐷) = (Hom ‘𝐷)
8 simprl 808 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (𝑉 ∩ Cat))
91, 2, 5catcbas 16960 . . . . . . . 8 (𝜑 → (Base‘𝐷) = (𝑉 ∩ Cat))
109adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (Base‘𝐷) = (𝑉 ∩ Cat))
118, 10eleqtrrd 2851 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐷))
12 simprr 810 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (𝑉 ∩ Cat))
1312, 10eleqtrrd 2851 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐷))
141, 2, 6, 7, 11, 13catchom 16962 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥 Func 𝑦))
15 resscatc.c . . . . . 6 𝐶 = (CatCat‘𝑈)
16 eqid 2769 . . . . . 6 (Base‘𝐶) = (Base‘𝐶)
173adantr 473 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑈𝑊)
18 eqid 2769 . . . . . 6 (Hom ‘𝐶) = (Hom ‘𝐶)
1915, 16, 3catcbas 16960 . . . . . . . . . . . 12 (𝜑 → (Base‘𝐶) = (𝑈 ∩ Cat))
2019ineq2d 3962 . . . . . . . . . . 11 (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (𝑉 ∩ (𝑈 ∩ Cat)))
21 inass 3969 . . . . . . . . . . 11 ((𝑉𝑈) ∩ Cat) = (𝑉 ∩ (𝑈 ∩ Cat))
2220, 21syl6reqr 2822 . . . . . . . . . 10 (𝜑 → ((𝑉𝑈) ∩ Cat) = (𝑉 ∩ (Base‘𝐶)))
23 df-ss 3734 . . . . . . . . . . . 12 (𝑉𝑈 ↔ (𝑉𝑈) = 𝑉)
244, 23sylib 208 . . . . . . . . . . 11 (𝜑 → (𝑉𝑈) = 𝑉)
2524ineq1d 3961 . . . . . . . . . 10 (𝜑 → ((𝑉𝑈) ∩ Cat) = (𝑉 ∩ Cat))
26 eqid 2769 . . . . . . . . . . . 12 (𝐶s 𝑉) = (𝐶s 𝑉)
2726, 16ressbas 16143 . . . . . . . . . . 11 (𝑉 ∈ V → (𝑉 ∩ (Base‘𝐶)) = (Base‘(𝐶s 𝑉)))
285, 27syl 17 . . . . . . . . . 10 (𝜑 → (𝑉 ∩ (Base‘𝐶)) = (Base‘(𝐶s 𝑉)))
2922, 25, 283eqtr3d 2811 . . . . . . . . 9 (𝜑 → (𝑉 ∩ Cat) = (Base‘(𝐶s 𝑉)))
3026, 16ressbasss 16145 . . . . . . . . 9 (Base‘(𝐶s 𝑉)) ⊆ (Base‘𝐶)
3129, 30syl6eqss 3801 . . . . . . . 8 (𝜑 → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
3231adantr 473 . . . . . . 7 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
3332, 8sseldd 3750 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑥 ∈ (Base‘𝐶))
3432, 12sseldd 3750 . . . . . 6 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → 𝑦 ∈ (Base‘𝐶))
3515, 16, 17, 18, 33, 34catchom 16962 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥 Func 𝑦))
3626, 18resshom 16292 . . . . . . 7 (𝑉 ∈ V → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
375, 36syl 17 . . . . . 6 (𝜑 → (Hom ‘𝐶) = (Hom ‘(𝐶s 𝑉)))
3837oveqdr 6817 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘(𝐶s 𝑉))𝑦))
3914, 35, 383eqtr2rd 2810 . . . 4 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) → (𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
4039ralrimivva 3118 . . 3 (𝜑 → ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦))
41 eqid 2769 . . . 4 (Hom ‘(𝐶s 𝑉)) = (Hom ‘(𝐶s 𝑉))
429eqcomd 2775 . . . 4 (𝜑 → (𝑉 ∩ Cat) = (Base‘𝐷))
4341, 7, 29, 42homfeq 16567 . . 3 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ↔ ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)(𝑥(Hom ‘(𝐶s 𝑉))𝑦) = (𝑥(Hom ‘𝐷)𝑦)))
4440, 43mpbird 247 . 2 (𝜑 → (Homf ‘(𝐶s 𝑉)) = (Homf𝐷))
455ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑉 ∈ V)
46 eqid 2769 . . . . . . . 8 (comp‘𝐷) = (comp‘𝐷)
47 simplr1 1258 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ (𝑉 ∩ Cat))
489ad2antrr 761 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (Base‘𝐷) = (𝑉 ∩ Cat))
4947, 48eleqtrrd 2851 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ (Base‘𝐷))
50 simplr2 1260 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ (𝑉 ∩ Cat))
5150, 48eleqtrrd 2851 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ (Base‘𝐷))
52 simplr3 1262 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ (𝑉 ∩ Cat))
5352, 48eleqtrrd 2851 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ (Base‘𝐷))
54 simprl 808 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦))
551, 2, 45, 7, 49, 51catchom 16962 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥 Func 𝑦))
5654, 55eleqtrd 2850 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑓 ∈ (𝑥 Func 𝑦))
57 simprr 810 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))
581, 2, 45, 7, 51, 53catchom 16962 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑦(Hom ‘𝐷)𝑧) = (𝑦 Func 𝑧))
5957, 58eleqtrd 2850 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑔 ∈ (𝑦 Func 𝑧))
601, 2, 45, 46, 49, 51, 53, 56, 59catcco 16964 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔func 𝑓))
613ad2antrr 761 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑈𝑊)
62 eqid 2769 . . . . . . . 8 (comp‘𝐶) = (comp‘𝐶)
6331ad2antrr 761 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑉 ∩ Cat) ⊆ (Base‘𝐶))
6463, 47sseldd 3750 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑥 ∈ (Base‘𝐶))
6563, 50sseldd 3750 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑦 ∈ (Base‘𝐶))
6663, 52sseldd 3750 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → 𝑧 ∈ (Base‘𝐶))
6715, 16, 61, 62, 64, 65, 66, 56, 59catcco 16964 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔func 𝑓))
6826, 62ressco 16293 . . . . . . . . . . 11 (𝑉 ∈ V → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
695, 68syl 17 . . . . . . . . . 10 (𝜑 → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
7069ad2antrr 761 . . . . . . . . 9 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (comp‘𝐶) = (comp‘(𝐶s 𝑉)))
7170oveqd 6808 . . . . . . . 8 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧))
7271oveqd 6808 . . . . . . 7 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
7360, 67, 723eqtr2d 2809 . . . . . 6 (((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦) ∧ 𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
7473ralrimivva 3118 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) → ∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
7574ralrimivvva 3119 . . . 4 (𝜑 → ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)∀𝑧 ∈ (𝑉 ∩ Cat)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓))
76 eqid 2769 . . . . 5 (comp‘(𝐶s 𝑉)) = (comp‘(𝐶s 𝑉))
7744eqcomd 2775 . . . . 5 (𝜑 → (Homf𝐷) = (Homf ‘(𝐶s 𝑉)))
7846, 76, 7, 42, 29, 77comfeq 16579 . . . 4 (𝜑 → ((compf𝐷) = (compf‘(𝐶s 𝑉)) ↔ ∀𝑥 ∈ (𝑉 ∩ Cat)∀𝑦 ∈ (𝑉 ∩ Cat)∀𝑧 ∈ (𝑉 ∩ Cat)∀𝑓 ∈ (𝑥(Hom ‘𝐷)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐷)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘(𝐶s 𝑉))𝑧)𝑓)))
7975, 78mpbird 247 . . 3 (𝜑 → (compf𝐷) = (compf‘(𝐶s 𝑉)))
8079eqcomd 2775 . 2 (𝜑 → (compf‘(𝐶s 𝑉)) = (compf𝐷))
8144, 80jca 556 1 (𝜑 → ((Homf ‘(𝐶s 𝑉)) = (Homf𝐷) ∧ (compf‘(𝐶s 𝑉)) = (compf𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1069   = wceq 1629  wcel 2143  wral 3059  Vcvv 3348  cin 3719  wss 3720  cop 4319  cfv 6030  (class class class)co 6791  Basecbs 16070  s cress 16071  Hom chom 16166  compcco 16167  Catccat 16538  Homf chomf 16540  compfccomf 16541   Func cfunc 16727  func ccofu 16729  CatCatccatc 16957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094  ax-cnex 10192  ax-resscn 10193  ax-1cn 10194  ax-icn 10195  ax-addcl 10196  ax-addrcl 10197  ax-mulcl 10198  ax-mulrcl 10199  ax-mulcom 10200  ax-addass 10201  ax-mulass 10202  ax-distr 10203  ax-i2m1 10204  ax-1ne0 10205  ax-1rid 10206  ax-rnegex 10207  ax-rrecex 10208  ax-cnre 10209  ax-pre-lttri 10210  ax-pre-lttrn 10211  ax-pre-ltadd 10212  ax-pre-mulgt0 10213
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1070  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-nel 3045  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-pss 3736  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4572  df-int 4609  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-tr 4884  df-id 5156  df-eprel 5161  df-po 5169  df-so 5170  df-fr 5207  df-we 5209  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6752  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-om 7211  df-1st 7313  df-2nd 7314  df-wrecs 7557  df-recs 7619  df-rdg 7657  df-1o 7711  df-oadd 7715  df-er 7894  df-en 8108  df-dom 8109  df-sdom 8110  df-fin 8111  df-pnf 10276  df-mnf 10277  df-xr 10278  df-ltxr 10279  df-le 10280  df-sub 10468  df-neg 10469  df-nn 11221  df-2 11279  df-3 11280  df-4 11281  df-5 11282  df-6 11283  df-7 11284  df-8 11285  df-9 11286  df-n0 11493  df-z 11578  df-dec 11694  df-uz 11888  df-fz 12533  df-struct 16072  df-ndx 16073  df-slot 16074  df-base 16076  df-sets 16077  df-ress 16078  df-hom 16180  df-cco 16181  df-homf 16544  df-comf 16545  df-catc 16958
This theorem is referenced by: (None)
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