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Theorem resssetc 17983
Description: The restriction of the category of sets to a subset is the category of sets in the subset. Thus, the SetCatβ€˜π‘ˆ categories for different π‘ˆ are full subcategories of each other. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resssetc.c 𝐢 = (SetCatβ€˜π‘ˆ)
resssetc.d 𝐷 = (SetCatβ€˜π‘‰)
resssetc.1 (πœ‘ β†’ π‘ˆ ∈ π‘Š)
resssetc.2 (πœ‘ β†’ 𝑉 βŠ† π‘ˆ)
Assertion
Ref Expression
resssetc (πœ‘ β†’ ((Homf β€˜(𝐢 β†Ύs 𝑉)) = (Homf β€˜π·) ∧ (compfβ€˜(𝐢 β†Ύs 𝑉)) = (compfβ€˜π·)))

Proof of Theorem resssetc
Dummy variables 𝑓 𝑔 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resssetc.d . . . . . 6 𝐷 = (SetCatβ€˜π‘‰)
2 resssetc.1 . . . . . . . 8 (πœ‘ β†’ π‘ˆ ∈ π‘Š)
3 resssetc.2 . . . . . . . 8 (πœ‘ β†’ 𝑉 βŠ† π‘ˆ)
42, 3ssexd 5282 . . . . . . 7 (πœ‘ β†’ 𝑉 ∈ V)
54adantr 482 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑉 ∈ V)
6 eqid 2733 . . . . . 6 (Hom β€˜π·) = (Hom β€˜π·)
7 simprl 770 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ 𝑉)
8 simprr 772 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑦 ∈ 𝑉)
91, 5, 6, 7, 8setchom 17971 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯(Hom β€˜π·)𝑦) = (𝑦 ↑m π‘₯))
10 resssetc.c . . . . . 6 𝐢 = (SetCatβ€˜π‘ˆ)
112adantr 482 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ π‘ˆ ∈ π‘Š)
12 eqid 2733 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
133adantr 482 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑉 βŠ† π‘ˆ)
1413, 7sseldd 3946 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ π‘ˆ)
1513, 8sseldd 3946 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑦 ∈ π‘ˆ)
1610, 11, 12, 14, 15setchom 17971 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯(Hom β€˜πΆ)𝑦) = (𝑦 ↑m π‘₯))
17 eqid 2733 . . . . . . . 8 (𝐢 β†Ύs 𝑉) = (𝐢 β†Ύs 𝑉)
1817, 12resshom 17305 . . . . . . 7 (𝑉 ∈ V β†’ (Hom β€˜πΆ) = (Hom β€˜(𝐢 β†Ύs 𝑉)))
194, 18syl 17 . . . . . 6 (πœ‘ β†’ (Hom β€˜πΆ) = (Hom β€˜(𝐢 β†Ύs 𝑉)))
2019oveqdr 7386 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯(Hom β€˜πΆ)𝑦) = (π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦))
219, 16, 203eqtr2rd 2780 . . . 4 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦) = (π‘₯(Hom β€˜π·)𝑦))
2221ralrimivva 3194 . . 3 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦) = (π‘₯(Hom β€˜π·)𝑦))
23 eqid 2733 . . . 4 (Hom β€˜(𝐢 β†Ύs 𝑉)) = (Hom β€˜(𝐢 β†Ύs 𝑉))
2410, 2setcbas 17969 . . . . . 6 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΆ))
253, 24sseqtrd 3985 . . . . 5 (πœ‘ β†’ 𝑉 βŠ† (Baseβ€˜πΆ))
26 eqid 2733 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
2717, 26ressbas2 17125 . . . . 5 (𝑉 βŠ† (Baseβ€˜πΆ) β†’ 𝑉 = (Baseβ€˜(𝐢 β†Ύs 𝑉)))
2825, 27syl 17 . . . 4 (πœ‘ β†’ 𝑉 = (Baseβ€˜(𝐢 β†Ύs 𝑉)))
291, 4setcbas 17969 . . . 4 (πœ‘ β†’ 𝑉 = (Baseβ€˜π·))
3023, 6, 28, 29homfeq 17579 . . 3 (πœ‘ β†’ ((Homf β€˜(𝐢 β†Ύs 𝑉)) = (Homf β€˜π·) ↔ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 (π‘₯(Hom β€˜(𝐢 β†Ύs 𝑉))𝑦) = (π‘₯(Hom β€˜π·)𝑦)))
3122, 30mpbird 257 . 2 (πœ‘ β†’ (Homf β€˜(𝐢 β†Ύs 𝑉)) = (Homf β€˜π·))
324ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑉 ∈ V)
33 eqid 2733 . . . . . . . 8 (compβ€˜π·) = (compβ€˜π·)
34 simplr1 1216 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ π‘₯ ∈ 𝑉)
35 simplr2 1217 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑦 ∈ 𝑉)
36 simplr3 1218 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑧 ∈ 𝑉)
37 simprl 770 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦))
381, 32, 6, 34, 35elsetchom 17972 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ↔ 𝑓:π‘₯βŸΆπ‘¦))
3937, 38mpbid 231 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑓:π‘₯βŸΆπ‘¦)
40 simprr 772 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))
411, 32, 6, 35, 36elsetchom 17972 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑔 ∈ (𝑦(Hom β€˜π·)𝑧) ↔ 𝑔:π‘¦βŸΆπ‘§))
4240, 41mpbid 231 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑔:π‘¦βŸΆπ‘§)
431, 32, 33, 34, 35, 36, 39, 42setcco 17974 . . . . . . 7 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜π·)𝑧)𝑓) = (𝑔 ∘ 𝑓))
442ad2antrr 725 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ π‘ˆ ∈ π‘Š)
45 eqid 2733 . . . . . . . 8 (compβ€˜πΆ) = (compβ€˜πΆ)
463ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑉 βŠ† π‘ˆ)
4746, 34sseldd 3946 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ π‘₯ ∈ π‘ˆ)
4846, 35sseldd 3946 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑦 ∈ π‘ˆ)
4946, 36sseldd 3946 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ 𝑧 ∈ π‘ˆ)
5010, 44, 45, 47, 48, 49, 39, 42setcco 17974 . . . . . . 7 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (𝑔(⟨π‘₯, π‘¦βŸ©(compβ€˜πΆ)𝑧)𝑓) = (𝑔 ∘ 𝑓))
5117, 45ressco 17306 . . . . . . . . . . 11 (𝑉 ∈ V β†’ (compβ€˜πΆ) = (compβ€˜(𝐢 β†Ύs 𝑉)))
524, 51syl 17 . . . . . . . . . 10 (πœ‘ β†’ (compβ€˜πΆ) = (compβ€˜(𝐢 β†Ύs 𝑉)))
5352ad2antrr 725 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (𝑓 ∈ (π‘₯(Hom β€˜π·)𝑦) ∧ 𝑔 ∈ (𝑦(Hom β€˜π·)𝑧))) β†’ (compβ€˜πΆ) = (compβ€˜(𝐢 β†Ύs 𝑉)))
5453oveqd 7375 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (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))
5554oveqd 7375 . . . . . . 7 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (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))
5643, 50, 553eqtr2d 2779 . . . . . 6 (((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) ∧ (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))
5756ralrimivva 3194 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) 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))
5857ralrimivvva 3197 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 βˆ€π‘§ ∈ 𝑉 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))
59 eqid 2733 . . . . 5 (compβ€˜(𝐢 β†Ύs 𝑉)) = (compβ€˜(𝐢 β†Ύs 𝑉))
6031eqcomd 2739 . . . . 5 (πœ‘ β†’ (Homf β€˜π·) = (Homf β€˜(𝐢 β†Ύs 𝑉)))
6133, 59, 6, 29, 28, 60comfeq 17591 . . . 4 (πœ‘ β†’ ((compfβ€˜π·) = (compfβ€˜(𝐢 β†Ύs 𝑉)) ↔ βˆ€π‘₯ ∈ 𝑉 βˆ€π‘¦ ∈ 𝑉 βˆ€π‘§ ∈ 𝑉 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)))
6258, 61mpbird 257 . . 3 (πœ‘ β†’ (compfβ€˜π·) = (compfβ€˜(𝐢 β†Ύs 𝑉)))
6362eqcomd 2739 . 2 (πœ‘ β†’ (compfβ€˜(𝐢 β†Ύs 𝑉)) = (compfβ€˜π·))
6431, 63jca 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   βŠ† wss 3911  βŸ¨cop 4593   ∘ ccom 5638  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8768  Basecbs 17088   β†Ύs cress 17117  Hom chom 17149  compcco 17150  Homf chomf 17551  compfccomf 17552  SetCatcsetc 17966
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-map 8770  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-setc 17967
This theorem is referenced by:  funcsetcres2  17984
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