MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resscatc Structured version   Visualization version   GIF version

Theorem resscatc 18000
Description: The restriction of the category of categories to a subset is the category of categories in the subset. Thus, the CatCatβ€˜π‘ˆ categories for different π‘ˆ are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resscatc.c 𝐢 = (CatCatβ€˜π‘ˆ)
resscatc.d 𝐷 = (CatCatβ€˜π‘‰)
resscatc.1 (πœ‘ β†’ π‘ˆ ∈ π‘Š)
resscatc.2 (πœ‘ β†’ 𝑉 βŠ† π‘ˆ)
Assertion
Ref Expression
resscatc (πœ‘ β†’ ((Homf β€˜(𝐢 β†Ύs 𝑉)) = (Homf β€˜π·) ∧ (compfβ€˜(𝐢 β†Ύs 𝑉)) = (compfβ€˜π·)))

Proof of Theorem resscatc
Dummy variables 𝑓 𝑔 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resscatc.d . . . . . 6 𝐷 = (CatCatβ€˜π‘‰)
2 eqid 2733 . . . . . 6 (Baseβ€˜π·) = (Baseβ€˜π·)
3 resscatc.1 . . . . . . . 8 (πœ‘ β†’ π‘ˆ ∈ π‘Š)
4 resscatc.2 . . . . . . . 8 (πœ‘ β†’ 𝑉 βŠ† π‘ˆ)
53, 4ssexd 5282 . . . . . . 7 (πœ‘ β†’ 𝑉 ∈ V)
65adantr 482 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ 𝑉 ∈ V)
7 eqid 2733 . . . . . 6 (Hom β€˜π·) = (Hom β€˜π·)
8 simprl 770 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ π‘₯ ∈ (𝑉 ∩ Cat))
91, 2, 5catcbas 17992 . . . . . . . 8 (πœ‘ β†’ (Baseβ€˜π·) = (𝑉 ∩ Cat))
109adantr 482 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ (Baseβ€˜π·) = (𝑉 ∩ Cat))
118, 10eleqtrrd 2837 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ π‘₯ ∈ (Baseβ€˜π·))
12 simprr 772 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ 𝑦 ∈ (𝑉 ∩ Cat))
1312, 10eleqtrrd 2837 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ 𝑦 ∈ (Baseβ€˜π·))
141, 2, 6, 7, 11, 13catchom 17994 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ (π‘₯(Hom β€˜π·)𝑦) = (π‘₯ Func 𝑦))
15 resscatc.c . . . . . 6 𝐢 = (CatCatβ€˜π‘ˆ)
16 eqid 2733 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
173adantr 482 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ π‘ˆ ∈ π‘Š)
18 eqid 2733 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
19 inass 4180 . . . . . . . . . . 11 ((𝑉 ∩ π‘ˆ) ∩ Cat) = (𝑉 ∩ (π‘ˆ ∩ Cat))
2015, 16, 3catcbas 17992 . . . . . . . . . . . 12 (πœ‘ β†’ (Baseβ€˜πΆ) = (π‘ˆ ∩ Cat))
2120ineq2d 4173 . . . . . . . . . . 11 (πœ‘ β†’ (𝑉 ∩ (Baseβ€˜πΆ)) = (𝑉 ∩ (π‘ˆ ∩ Cat)))
2219, 21eqtr4id 2792 . . . . . . . . . 10 (πœ‘ β†’ ((𝑉 ∩ π‘ˆ) ∩ Cat) = (𝑉 ∩ (Baseβ€˜πΆ)))
23 df-ss 3928 . . . . . . . . . . . 12 (𝑉 βŠ† π‘ˆ ↔ (𝑉 ∩ π‘ˆ) = 𝑉)
244, 23sylib 217 . . . . . . . . . . 11 (πœ‘ β†’ (𝑉 ∩ π‘ˆ) = 𝑉)
2524ineq1d 4172 . . . . . . . . . 10 (πœ‘ β†’ ((𝑉 ∩ π‘ˆ) ∩ Cat) = (𝑉 ∩ Cat))
26 eqid 2733 . . . . . . . . . . . 12 (𝐢 β†Ύs 𝑉) = (𝐢 β†Ύs 𝑉)
2726, 16ressbas 17123 . . . . . . . . . . 11 (𝑉 ∈ V β†’ (𝑉 ∩ (Baseβ€˜πΆ)) = (Baseβ€˜(𝐢 β†Ύs 𝑉)))
285, 27syl 17 . . . . . . . . . 10 (πœ‘ β†’ (𝑉 ∩ (Baseβ€˜πΆ)) = (Baseβ€˜(𝐢 β†Ύs 𝑉)))
2922, 25, 283eqtr3d 2781 . . . . . . . . 9 (πœ‘ β†’ (𝑉 ∩ Cat) = (Baseβ€˜(𝐢 β†Ύs 𝑉)))
3026, 16ressbasss 17126 . . . . . . . . 9 (Baseβ€˜(𝐢 β†Ύs 𝑉)) βŠ† (Baseβ€˜πΆ)
3129, 30eqsstrdi 3999 . . . . . . . 8 (πœ‘ β†’ (𝑉 ∩ Cat) βŠ† (Baseβ€˜πΆ))
3231adantr 482 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ (𝑉 ∩ Cat) βŠ† (Baseβ€˜πΆ))
3332, 8sseldd 3946 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ π‘₯ ∈ (Baseβ€˜πΆ))
3432, 12sseldd 3946 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
3515, 16, 17, 18, 33, 34catchom 17994 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ (π‘₯(Hom β€˜πΆ)𝑦) = (π‘₯ Func 𝑦))
3626, 18resshom 17305 . . . . . . 7 (𝑉 ∈ V β†’ (Hom β€˜πΆ) = (Hom β€˜(𝐢 β†Ύs 𝑉)))
375, 36syl 17 . . . . . 6 (πœ‘ β†’ (Hom β€˜πΆ) = (Hom β€˜(𝐢 β†Ύs 𝑉)))
3837oveqdr 7386 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ (π‘₯(Hom β€˜πΆ)𝑦) = (π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦))
3914, 35, 383eqtr2rd 2780 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat))) β†’ (π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦) = (π‘₯(Hom β€˜π·)𝑦))
4039ralrimivva 3194 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ (𝑉 ∩ Cat)βˆ€π‘¦ ∈ (𝑉 ∩ Cat)(π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦) = (π‘₯(Hom β€˜π·)𝑦))
41 eqid 2733 . . . 4 (Hom β€˜(𝐢 β†Ύs 𝑉)) = (Hom β€˜(𝐢 β†Ύs 𝑉))
429eqcomd 2739 . . . 4 (πœ‘ β†’ (𝑉 ∩ Cat) = (Baseβ€˜π·))
4341, 7, 29, 42homfeq 17579 . . 3 (πœ‘ β†’ ((Homf β€˜(𝐢 β†Ύs 𝑉)) = (Homf β€˜π·) ↔ βˆ€π‘₯ ∈ (𝑉 ∩ Cat)βˆ€π‘¦ ∈ (𝑉 ∩ Cat)(π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦) = (π‘₯(Hom β€˜π·)𝑦)))
4440, 43mpbird 257 . 2 (πœ‘ β†’ (Homf β€˜(𝐢 β†Ύs 𝑉)) = (Homf β€˜π·))
455ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑉 ∈ V)
46 eqid 2733 . . . . . . . 8 (compβ€˜π·) = (compβ€˜π·)
47 simplr1 1216 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ π‘₯ ∈ (𝑉 ∩ Cat))
489ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (Baseβ€˜π·) = (𝑉 ∩ Cat))
4947, 48eleqtrrd 2837 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ π‘₯ ∈ (Baseβ€˜π·))
50 simplr2 1217 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑦 ∈ (𝑉 ∩ Cat))
5150, 48eleqtrrd 2837 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑦 ∈ (Baseβ€˜π·))
52 simplr3 1218 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑧 ∈ (𝑉 ∩ Cat))
5352, 48eleqtrrd 2837 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑧 ∈ (Baseβ€˜π·))
54 simprl 770 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦))
551, 2, 45, 7, 49, 51catchom 17994 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (π‘₯(Hom β€˜π·)𝑦) = (π‘₯ Func 𝑦))
5654, 55eleqtrd 2836 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑓 ∈ (π‘₯ Func 𝑦))
57 simprr 772 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))
581, 2, 45, 7, 51, 53catchom 17994 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑦(Hom β€˜π·)𝑧) = (𝑦 Func 𝑧))
5957, 58eleqtrd 2836 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑔 ∈ (𝑦 Func 𝑧))
601, 2, 45, 46, 49, 51, 53, 56, 59catcco 17996 . . . . . . 7 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π·)𝑧)𝑓) = (𝑔 ∘func 𝑓))
613ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ π‘ˆ ∈ π‘Š)
62 eqid 2733 . . . . . . . 8 (compβ€˜πΆ) = (compβ€˜πΆ)
6331ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑉 ∩ Cat) βŠ† (Baseβ€˜πΆ))
6463, 47sseldd 3946 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ π‘₯ ∈ (Baseβ€˜πΆ))
6563, 50sseldd 3946 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
6663, 52sseldd 3946 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑧 ∈ (Baseβ€˜πΆ))
6715, 16, 61, 62, 64, 65, 66, 56, 59catcco 17996 . . . . . . 7 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜πΆ)𝑧)𝑓) = (𝑔 ∘func 𝑓))
6826, 62ressco 17306 . . . . . . . . . . 11 (𝑉 ∈ V β†’ (compβ€˜πΆ) = (compβ€˜(𝐢 β†Ύs 𝑉)))
695, 68syl 17 . . . . . . . . . 10 (πœ‘ β†’ (compβ€˜πΆ) = (compβ€˜(𝐢 β†Ύs 𝑉)))
7069ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (compβ€˜πΆ) = (compβ€˜(𝐢 β†Ύs 𝑉)))
7170oveqd 7375 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (⟨π‘₯, π‘¦βŸ©(compβ€˜πΆ)𝑧) = (⟨π‘₯, π‘¦βŸ©(compβ€˜(𝐢 β†Ύs 𝑉))𝑧))
7271oveqd 7375 . . . . . . 7 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜πΆ)𝑧)𝑓) = (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝐢 β†Ύs 𝑉))𝑧)𝑓))
7360, 67, 723eqtr2d 2779 . . . . . 6 (((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π·)𝑧)𝑓) = (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝐢 β†Ύs 𝑉))𝑧)𝑓))
7473ralrimivva 3194 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ (𝑉 ∩ Cat) ∧ 𝑦 ∈ (𝑉 ∩ Cat) ∧ 𝑧 ∈ (𝑉 ∩ Cat))) β†’ βˆ€π‘“ ∈ (π‘₯(Hom β€˜π·)𝑦)βˆ€π‘” ∈ (𝑦(Hom β€˜π·)𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π·)𝑧)𝑓) = (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝐢 β†Ύs 𝑉))𝑧)𝑓))
7574ralrimivvva 3197 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ (𝑉 ∩ Cat)βˆ€π‘¦ ∈ (𝑉 ∩ Cat)βˆ€π‘§ ∈ (𝑉 ∩ Cat)βˆ€π‘“ ∈ (π‘₯(Hom β€˜π·)𝑦)βˆ€π‘” ∈ (𝑦(Hom β€˜π·)𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π·)𝑧)𝑓) = (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝐢 β†Ύs 𝑉))𝑧)𝑓))
76 eqid 2733 . . . . 5 (compβ€˜(𝐢 β†Ύs 𝑉)) = (compβ€˜(𝐢 β†Ύs 𝑉))
7744eqcomd 2739 . . . . 5 (πœ‘ β†’ (Homf β€˜π·) = (Homf β€˜(𝐢 β†Ύs 𝑉)))
7846, 76, 7, 42, 29, 77comfeq 17591 . . . 4 (πœ‘ β†’ ((compfβ€˜π·) = (compfβ€˜(𝐢 β†Ύs 𝑉)) ↔ βˆ€π‘₯ ∈ (𝑉 ∩ Cat)βˆ€π‘¦ ∈ (𝑉 ∩ Cat)βˆ€π‘§ ∈ (𝑉 ∩ Cat)βˆ€π‘“ ∈ (π‘₯(Hom β€˜π·)𝑦)βˆ€π‘” ∈ (𝑦(Hom β€˜π·)𝑧)(𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π·)𝑧)𝑓) = (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜(𝐢 β†Ύs 𝑉))𝑧)𝑓)))
7975, 78mpbird 257 . . 3 (πœ‘ β†’ (compfβ€˜π·) = (compfβ€˜(𝐢 β†Ύs 𝑉)))
8079eqcomd 2739 . 2 (πœ‘ β†’ (compfβ€˜(𝐢 β†Ύs 𝑉)) = (compfβ€˜π·))
8144, 80jca 513 1 (πœ‘ β†’ ((Homf β€˜(𝐢 β†Ύs 𝑉)) = (Homf β€˜π·) ∧ (compfβ€˜(𝐢 β†Ύs 𝑉)) = (compfβ€˜π·)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  βŸ¨cop 4593  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088   β†Ύs cress 17117  Hom chom 17149  compcco 17150  Catccat 17549  Homf chomf 17551  compfccomf 17552   Func cfunc 17745   ∘func ccofu 17747  CatCatccatc 17989
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-hom 17162  df-cco 17163  df-homf 17555  df-comf 17556  df-catc 17990
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator