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Theorem fthestrcsetc 18101
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 18100 . 2 (πœ‘ β†’ 𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 18098 . . . 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 481 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
11 eqid 2732 . . . . . . . . . . . . 13 (Hom β€˜πΈ) = (Hom β€˜πΈ)
121, 5estrcbas 18075 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΈ))
133, 12eqtr4id 2791 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐡 = π‘ˆ)
1413eleq2d 2819 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
1716impcom 408 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
1813eleq2d 2819 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑏 ∈ 𝐡 ↔ 𝑏 ∈ π‘ˆ))
1918biimpcd 248 . . . . . . . . . . . . . . 15 (𝑏 ∈ 𝐡 β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
2019adantl 482 . . . . . . . . . . . . . 14 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
2120impcom 408 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
22 eqid 2732 . . . . . . . . . . . . 13 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
23 eqid 2732 . . . . . . . . . . . . 13 (Baseβ€˜π‘) = (Baseβ€˜π‘)
241, 10, 11, 17, 21, 22, 23estrchom 18077 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜πΈ)𝑏) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
2524eleq2d 2819 . . . . . . . . . . 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 18096 . . . . . . . . . . . 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 1121 . . . . . . . . . . 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 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€˜β„Ž) = β„Ž))
3130impcom 408 . . . . . . 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 2819 . . . . . . . . . . 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 18096 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) ∧ π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜)
34333expia 1121 . . . . . . . . . . 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 482 . . . . . . . 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 408 . . . . . . 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 2748 . . . . . 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 3200 . . . 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 7253 . . . 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 583 . . 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 3200 . 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 2732 . . 3 (Hom β€˜π‘†) = (Hom β€˜π‘†)
463, 11, 45isfth2 17865 . 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 583 1 (πœ‘ β†’ 𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   class class class wbr 5148   ↦ cmpt 5231   I cid 5573   β†Ύ cres 5678  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410   ↑m cmap 8819  WUnicwun 10694  Basecbs 17143  Hom chom 17207   Func cfunc 17803   Faith cfth 17853  SetCatcsetc 18024  ExtStrCatcestrc 18072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-ixp 8891  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-wun 10696  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12472  df-z 12558  df-dec 12677  df-uz 12822  df-fz 13484  df-struct 17079  df-slot 17114  df-ndx 17126  df-base 17144  df-hom 17220  df-cco 17221  df-cat 17611  df-cid 17612  df-func 17807  df-fth 17855  df-setc 18025  df-estrc 18073
This theorem is referenced by:  equivestrcsetc  18103
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