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Theorem fullestrcsetc 18133
Description: The "natural forgetful functor" from the category of extensible structures into the category of sets which sends each extensible structure to its base set is full. (Contributed by AV, 2-Apr-2020.)
Hypotheses
Ref Expression
funcestrcsetc.e 𝐸 = (ExtStrCatβ€˜π‘ˆ)
funcestrcsetc.s 𝑆 = (SetCatβ€˜π‘ˆ)
funcestrcsetc.b 𝐡 = (Baseβ€˜πΈ)
funcestrcsetc.c 𝐢 = (Baseβ€˜π‘†)
funcestrcsetc.u (πœ‘ β†’ π‘ˆ ∈ WUni)
funcestrcsetc.f (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
funcestrcsetc.g (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
Assertion
Ref Expression
fullestrcsetc (πœ‘ β†’ 𝐹(𝐸 Full 𝑆)𝐺)
Distinct variable groups:   π‘₯,𝐡   πœ‘,π‘₯   π‘₯,𝐢   𝑦,𝐡,π‘₯   πœ‘,𝑦
Allowed substitution hints:   𝐢(𝑦)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐸(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)

Proof of Theorem fullestrcsetc
Dummy variables π‘Ž 𝑏 β„Ž π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcestrcsetc.e . . 3 𝐸 = (ExtStrCatβ€˜π‘ˆ)
2 funcestrcsetc.s . . 3 𝑆 = (SetCatβ€˜π‘ˆ)
3 funcestrcsetc.b . . 3 𝐡 = (Baseβ€˜πΈ)
4 funcestrcsetc.c . . 3 𝐢 = (Baseβ€˜π‘†)
5 funcestrcsetc.u . . 3 (πœ‘ β†’ π‘ˆ ∈ WUni)
6 funcestrcsetc.f . . 3 (πœ‘ β†’ 𝐹 = (π‘₯ ∈ 𝐡 ↦ (Baseβ€˜π‘₯)))
7 funcestrcsetc.g . . 3 (πœ‘ β†’ 𝐺 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ ( I β†Ύ ((Baseβ€˜π‘¦) ↑m (Baseβ€˜π‘₯)))))
81, 2, 3, 4, 5, 6, 7funcestrcsetc 18131 . 2 (πœ‘ β†’ 𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 18129 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)⟢((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)))
105adantr 480 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
11 eqid 2727 . . . . . . . 8 (Hom β€˜π‘†) = (Hom β€˜π‘†)
121, 2, 3, 4, 5, 6funcestrcsetclem2 18123 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (πΉβ€˜π‘Ž) ∈ π‘ˆ)
1312adantrr 716 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘Ž) ∈ π‘ˆ)
141, 2, 3, 4, 5, 6funcestrcsetclem2 18123 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (πΉβ€˜π‘) ∈ π‘ˆ)
1514adantrl 715 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘) ∈ π‘ˆ)
162, 10, 11, 13, 15elsetchom 18061 . . . . . . 7 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž ∈ ((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)) ↔ β„Ž:(πΉβ€˜π‘Ž)⟢(πΉβ€˜π‘)))
171, 2, 3, 4, 5, 6funcestrcsetclem1 18122 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ 𝐡) β†’ (πΉβ€˜π‘Ž) = (Baseβ€˜π‘Ž))
1817adantrr 716 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘Ž) = (Baseβ€˜π‘Ž))
191, 2, 3, 4, 5, 6funcestrcsetclem1 18122 . . . . . . . . 9 ((πœ‘ ∧ 𝑏 ∈ 𝐡) β†’ (πΉβ€˜π‘) = (Baseβ€˜π‘))
2019adantrl 715 . . . . . . . 8 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (πΉβ€˜π‘) = (Baseβ€˜π‘))
2118, 20feq23d 6711 . . . . . . 7 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž:(πΉβ€˜π‘Ž)⟢(πΉβ€˜π‘) ↔ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)))
2216, 21bitrd 279 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž ∈ ((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)) ↔ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)))
23 fvex 6904 . . . . . . . . . . . . 13 (Baseβ€˜π‘) ∈ V
24 fvex 6904 . . . . . . . . . . . . 13 (Baseβ€˜π‘Ž) ∈ V
2523, 24pm3.2i 470 . . . . . . . . . . . 12 ((Baseβ€˜π‘) ∈ V ∧ (Baseβ€˜π‘Ž) ∈ V)
26 elmapg 8849 . . . . . . . . . . . 12 (((Baseβ€˜π‘) ∈ V ∧ (Baseβ€˜π‘Ž) ∈ V) β†’ (β„Ž ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ↔ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)))
2725, 26mp1i 13 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) ↔ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)))
2827biimpar 477 . . . . . . . . . 10 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) β†’ β„Ž ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
29 equequ2 2022 . . . . . . . . . . 11 (π‘˜ = β„Ž β†’ (β„Ž = π‘˜ ↔ β„Ž = β„Ž))
3029adantl 481 . . . . . . . . . 10 ((((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) ∧ π‘˜ = β„Ž) β†’ (β„Ž = π‘˜ ↔ β„Ž = β„Ž))
31 eqidd 2728 . . . . . . . . . 10 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) β†’ β„Ž = β„Ž)
3228, 30, 31rspcedvd 3609 . . . . . . . . 9 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) β†’ βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = π‘˜)
33 eqid 2727 . . . . . . . . . . . . . 14 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
34 eqid 2727 . . . . . . . . . . . . . 14 (Baseβ€˜π‘) = (Baseβ€˜π‘)
351, 2, 3, 4, 5, 6, 7, 33, 34funcestrcsetclem6 18127 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) ∧ π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜)
36353expa 1116 . . . . . . . . . . . 12 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜)
3736eqeq2d 2738 . . . . . . . . . . 11 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†’ (β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜) ↔ β„Ž = π‘˜))
3837rexbidva 3171 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜) ↔ βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = π‘˜))
3938adantr 480 . . . . . . . . 9 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) β†’ (βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜) ↔ βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = π‘˜))
4032, 39mpbird 257 . . . . . . . 8 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) β†’ βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜))
41 eqid 2727 . . . . . . . . . . 11 (Hom β€˜πΈ) = (Hom β€˜πΈ)
421, 5estrcbas 18106 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΈ))
433, 42eqtr4id 2786 . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐡 = π‘ˆ)
4443eleq2d 2814 . . . . . . . . . . . . . 14 (πœ‘ β†’ (π‘Ž ∈ 𝐡 ↔ π‘Ž ∈ π‘ˆ))
4544biimpcd 248 . . . . . . . . . . . . 13 (π‘Ž ∈ 𝐡 β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
4645adantr 480 . . . . . . . . . . . 12 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
4746impcom 407 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
4843eleq2d 2814 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝑏 ∈ 𝐡 ↔ 𝑏 ∈ π‘ˆ))
4948biimpcd 248 . . . . . . . . . . . . 13 (𝑏 ∈ 𝐡 β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
5049adantl 481 . . . . . . . . . . . 12 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
5150impcom 407 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
521, 10, 41, 47, 51, 33, 34estrchom 18108 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜πΈ)𝑏) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
5352rexeqdv 3321 . . . . . . . . 9 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (βˆƒπ‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜) ↔ βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜)))
5453adantr 480 . . . . . . . 8 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) β†’ (βˆƒπ‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜) ↔ βˆƒπ‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜)))
5540, 54mpbird 257 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘)) β†’ βˆƒπ‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜))
5655ex 412 . . . . . 6 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž:(Baseβ€˜π‘Ž)⟢(Baseβ€˜π‘) β†’ βˆƒπ‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜)))
5722, 56sylbid 239 . . . . 5 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž ∈ ((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)) β†’ βˆƒπ‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜)))
5857ralrimiv 3140 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ βˆ€β„Ž ∈ ((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘))βˆƒπ‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜))
59 dffo3 7106 . . . 4 ((π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–ontoβ†’((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)) ↔ ((π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)⟢((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)) ∧ βˆ€β„Ž ∈ ((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘))βˆƒπ‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)β„Ž = ((π‘ŽπΊπ‘)β€˜π‘˜)))
609, 58, 59sylanbrc 582 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–ontoβ†’((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)))
6160ralrimivva 3195 . 2 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–ontoβ†’((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)))
623, 11, 41isfull2 17891 . 2 (𝐹(𝐸 Full 𝑆)𝐺 ↔ (𝐹(𝐸 Func 𝑆)𝐺 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–ontoβ†’((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘))))
638, 61, 62sylanbrc 582 1 (πœ‘ β†’ 𝐹(𝐸 Full 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056  βˆƒwrex 3065  Vcvv 3469   class class class wbr 5142   ↦ cmpt 5225   I cid 5569   β†Ύ cres 5674  βŸΆwf 6538  β€“ontoβ†’wfo 6540  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416   ↑m cmap 8836  WUnicwun 10715  Basecbs 17171  Hom chom 17235   Func cfunc 17831   Full cful 17882  SetCatcsetc 18055  ExtStrCatcestrc 18103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8838  df-ixp 8908  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-wun 10717  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-3 12298  df-4 12299  df-5 12300  df-6 12301  df-7 12302  df-8 12303  df-9 12304  df-n0 12495  df-z 12581  df-dec 12700  df-uz 12845  df-fz 13509  df-struct 17107  df-slot 17142  df-ndx 17154  df-base 17172  df-hom 17248  df-cco 17249  df-cat 17639  df-cid 17640  df-func 17835  df-full 17884  df-setc 18056  df-estrc 18104
This theorem is referenced by:  equivestrcsetc  18134
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