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Theorem fthestrcsetc 18126
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 faithful. (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
fthestrcsetc (πœ‘ β†’ 𝐹(𝐸 Faith 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 fthestrcsetc
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 18125 . 2 (πœ‘ β†’ 𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 18123 . . . 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 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
11 eqid 2727 . . . . . . . . . . . . 13 (Hom β€˜πΈ) = (Hom β€˜πΈ)
121, 5estrcbas 18100 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΈ))
133, 12eqtr4id 2786 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐡 = π‘ˆ)
1413eleq2d 2814 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (π‘Ž ∈ 𝐡 ↔ π‘Ž ∈ π‘ˆ))
1514biimpcd 248 . . . . . . . . . . . . . . 15 (π‘Ž ∈ 𝐡 β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
1615adantr 480 . . . . . . . . . . . . . 14 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
1716impcom 407 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
1813eleq2d 2814 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑏 ∈ 𝐡 ↔ 𝑏 ∈ π‘ˆ))
1918biimpcd 248 . . . . . . . . . . . . . . 15 (𝑏 ∈ 𝐡 β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
2019adantl 481 . . . . . . . . . . . . . 14 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
2120impcom 407 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
22 eqid 2727 . . . . . . . . . . . . 13 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
23 eqid 2727 . . . . . . . . . . . . 13 (Baseβ€˜π‘) = (Baseβ€˜π‘)
241, 10, 11, 17, 21, 22, 23estrchom 18102 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜πΈ)𝑏) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
2524eleq2d 2814 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ↔ β„Ž ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
261, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 18121 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) ∧ β„Ž ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†’ ((π‘ŽπΊπ‘)β€˜β„Ž) = β„Ž)
27263expia 1119 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) β†’ ((π‘ŽπΊπ‘)β€˜β„Ž) = β„Ž))
2825, 27sylbid 239 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) β†’ ((π‘ŽπΊπ‘)β€˜β„Ž) = β„Ž))
2928com12 32 . . . . . . . . 9 (β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) β†’ ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((π‘ŽπΊπ‘)β€˜β„Ž) = β„Ž))
3029adantr 480 . . . . . . . 8 ((β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ∧ π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)) β†’ ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((π‘ŽπΊπ‘)β€˜β„Ž) = β„Ž))
3130impcom 407 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ (β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ∧ π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏))) β†’ ((π‘ŽπΊπ‘)β€˜β„Ž) = β„Ž)
3224eleq2d 2814 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ↔ π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))))
331, 2, 3, 4, 5, 6, 7, 22, 23funcestrcsetclem6 18121 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) ∧ π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜)
34333expia 1119 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜))
3532, 34sylbid 239 . . . . . . . . . 10 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜))
3635com12 32 . . . . . . . . 9 (π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏) β†’ ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜))
3736adantl 481 . . . . . . . 8 ((β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ∧ π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)) β†’ ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜))
3837impcom 407 . . . . . . 7 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ (β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ∧ π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏))) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜)
3931, 38eqeq12d 2743 . . . . . 6 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ (β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ∧ π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏))) β†’ (((π‘ŽπΊπ‘)β€˜β„Ž) = ((π‘ŽπΊπ‘)β€˜π‘˜) ↔ β„Ž = π‘˜))
4039biimpd 228 . . . . 5 (((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) ∧ (β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏) ∧ π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏))) β†’ (((π‘ŽπΊπ‘)β€˜β„Ž) = ((π‘ŽπΊπ‘)β€˜π‘˜) β†’ β„Ž = π‘˜))
4140ralrimivva 3195 . . . 4 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ βˆ€β„Ž ∈ (π‘Ž(Hom β€˜πΈ)𝑏)βˆ€π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)(((π‘ŽπΊπ‘)β€˜β„Ž) = ((π‘ŽπΊπ‘)β€˜π‘˜) β†’ β„Ž = π‘˜))
42 dff13 7259 . . . 4 ((π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–1-1β†’((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‘Ž(Hom β€˜πΈ)𝑏)βˆ€π‘˜ ∈ (π‘Ž(Hom β€˜πΈ)𝑏)(((π‘ŽπΊπ‘)β€˜β„Ž) = ((π‘ŽπΊπ‘)β€˜π‘˜) β†’ β„Ž = π‘˜)))
439, 41, 42sylanbrc 582 . . 3 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–1-1β†’((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)))
4443ralrimivva 3195 . 2 (πœ‘ β†’ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–1-1β†’((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘)))
45 eqid 2727 . . 3 (Hom β€˜π‘†) = (Hom β€˜π‘†)
463, 11, 45isfth2 17889 . 2 (𝐹(𝐸 Faith 𝑆)𝐺 ↔ (𝐹(𝐸 Func 𝑆)𝐺 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 (π‘ŽπΊπ‘):(π‘Ž(Hom β€˜πΈ)𝑏)–1-1β†’((πΉβ€˜π‘Ž)(Hom β€˜π‘†)(πΉβ€˜π‘))))
478, 44, 46sylanbrc 582 1 (πœ‘ β†’ 𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  βˆ€wral 3056   class class class wbr 5142   ↦ cmpt 5225   I cid 5569   β†Ύ cres 5674  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7414   ∈ cmpo 7416   ↑m cmap 8834  WUnicwun 10709  Basecbs 17165  Hom chom 17229   Func cfunc 17825   Faith cfth 17877  SetCatcsetc 18049  ExtStrCatcestrc 18097
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 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201
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 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-ixp 8906  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-wun 10711  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-fz 13503  df-struct 17101  df-slot 17136  df-ndx 17148  df-base 17166  df-hom 17242  df-cco 17243  df-cat 17633  df-cid 17634  df-func 17829  df-fth 17879  df-setc 18050  df-estrc 18098
This theorem is referenced by:  equivestrcsetc  18128
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