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Theorem resssetc 18054
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 5317 . . . . . . 7 (πœ‘ β†’ 𝑉 ∈ V)
54adantr 480 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑉 ∈ V)
6 eqid 2726 . . . . . 6 (Hom β€˜π·) = (Hom β€˜π·)
7 simprl 768 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ 𝑉)
8 simprr 770 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑦 ∈ 𝑉)
91, 5, 6, 7, 8setchom 18042 . . . . 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 480 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ π‘ˆ ∈ π‘Š)
12 eqid 2726 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
133adantr 480 . . . . . . 7 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑉 βŠ† π‘ˆ)
1413, 7sseldd 3978 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ π‘₯ ∈ π‘ˆ)
1513, 8sseldd 3978 . . . . . 6 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ 𝑦 ∈ π‘ˆ)
1610, 11, 12, 14, 15setchom 18042 . . . . 5 ((πœ‘ ∧ (π‘₯ ∈ 𝑉 ∧ 𝑦 ∈ 𝑉)) β†’ (π‘₯(Hom β€˜πΆ)𝑦) = (𝑦 ↑m π‘₯))
17 eqid 2726 . . . . . . . 8 (𝐢 β†Ύs 𝑉) = (𝐢 β†Ύs 𝑉)
1817, 12resshom 17373 . . . . . . 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 7433 . . . . 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 2773 . . . 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 2726 . . . 4 (Hom β€˜(𝐢 β†Ύs 𝑉)) = (Hom β€˜(𝐢 β†Ύs 𝑉))
2410, 2setcbas 18040 . . . . . 6 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΆ))
253, 24sseqtrd 4017 . . . . 5 (πœ‘ β†’ 𝑉 βŠ† (Baseβ€˜πΆ))
26 eqid 2726 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
2717, 26ressbas2 17191 . . . . 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 18040 . . . 4 (πœ‘ β†’ 𝑉 = (Baseβ€˜π·))
3023, 6, 28, 29homfeq 17647 . . 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 723 . . . . . . . 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 2726 . . . . . . . 8 (compβ€˜π·) = (compβ€˜π·)
34 simplr1 1212 . . . . . . . 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 1213 . . . . . . . 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 1214 . . . . . . . 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 768 . . . . . . . . 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 18043 . . . . . . . . 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 770 . . . . . . . . 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 18043 . . . . . . . . 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 18045 . . . . . . 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 723 . . . . . . . 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 2726 . . . . . . . 8 (compβ€˜πΆ) = (compβ€˜πΆ)
463ad2antrr 723 . . . . . . . . 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 3978 . . . . . . . 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 3978 . . . . . . . 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 3978 . . . . . . . 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 18045 . . . . . . 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 17374 . . . . . . . . . . 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 723 . . . . . . . . 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 7422 . . . . . . . 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 7422 . . . . . . 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 2772 . . . . . 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 2726 . . . . 5 (compβ€˜(𝐢 β†Ύs 𝑉)) = (compβ€˜(𝐢 β†Ύs 𝑉))
6031eqcomd 2732 . . . . 5 (πœ‘ β†’ (Homf β€˜π·) = (Homf β€˜(𝐢 β†Ύs 𝑉)))
6133, 59, 6, 29, 28, 60comfeq 17659 . . . 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 2732 . 2 (πœ‘ β†’ (compfβ€˜(𝐢 β†Ύs 𝑉)) = (compfβ€˜π·))
6431, 63jca 511 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βŠ† wss 3943  βŸ¨cop 4629   ∘ ccom 5673  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ↑m cmap 8822  Basecbs 17153   β†Ύs cress 17182  Hom chom 17217  compcco 17218  Homf chomf 17619  compfccomf 17620  SetCatcsetc 18037
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-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-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-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-hom 17230  df-cco 17231  df-homf 17623  df-comf 17624  df-setc 18038
This theorem is referenced by:  funcsetcres2  18055
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