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Theorem fthestrcsetc 18043
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 18042 . 2 (πœ‘ β†’ 𝐹(𝐸 Func 𝑆)𝐺)
91, 2, 3, 4, 5, 6, 7funcestrcsetclem8 18040 . . . 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 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘ˆ ∈ WUni)
11 eqid 2733 . . . . . . . . . . . . 13 (Hom β€˜πΈ) = (Hom β€˜πΈ)
121, 5estrcbas 18017 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ π‘ˆ = (Baseβ€˜πΈ))
133, 12eqtr4id 2792 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ 𝐡 = π‘ˆ)
1413eleq2d 2820 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (π‘Ž ∈ 𝐡 ↔ π‘Ž ∈ π‘ˆ))
1514biimpcd 249 . . . . . . . . . . . . . . 15 (π‘Ž ∈ 𝐡 β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
1615adantr 482 . . . . . . . . . . . . . 14 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ π‘Ž ∈ π‘ˆ))
1716impcom 409 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ π‘Ž ∈ π‘ˆ)
1813eleq2d 2820 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (𝑏 ∈ 𝐡 ↔ 𝑏 ∈ π‘ˆ))
1918biimpcd 249 . . . . . . . . . . . . . . 15 (𝑏 ∈ 𝐡 β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
2019adantl 483 . . . . . . . . . . . . . 14 ((π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) β†’ (πœ‘ β†’ 𝑏 ∈ π‘ˆ))
2120impcom 409 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ 𝑏 ∈ π‘ˆ)
22 eqid 2733 . . . . . . . . . . . . 13 (Baseβ€˜π‘Ž) = (Baseβ€˜π‘Ž)
23 eqid 2733 . . . . . . . . . . . . 13 (Baseβ€˜π‘) = (Baseβ€˜π‘)
241, 10, 11, 17, 21, 22, 23estrchom 18019 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡)) β†’ (π‘Ž(Hom β€˜πΈ)𝑏) = ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž)))
2524eleq2d 2820 . . . . . . . . . . 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 18038 . . . . . . . . . . . 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 1122 . . . . . . . . . . 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 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€˜β„Ž) = β„Ž))
3130impcom 409 . . . . . . 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 2820 . . . . . . . . . . 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 18038 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘Ž ∈ 𝐡 ∧ 𝑏 ∈ 𝐡) ∧ π‘˜ ∈ ((Baseβ€˜π‘) ↑m (Baseβ€˜π‘Ž))) β†’ ((π‘ŽπΊπ‘)β€˜π‘˜) = π‘˜)
34333expia 1122 . . . . . . . . . . 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 483 . . . . . . . 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 409 . . . . . . 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 2749 . . . . . 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 3194 . . . 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 7203 . . . 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 584 . . 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 3194 . 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 2733 . . 3 (Hom β€˜π‘†) = (Hom β€˜π‘†)
463, 11, 45isfth2 17807 . 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 584 1 (πœ‘ β†’ 𝐹(𝐸 Faith 𝑆)𝐺)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   class class class wbr 5106   ↦ cmpt 5189   I cid 5531   β†Ύ cres 5636  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360   ↑m cmap 8768  WUnicwun 10641  Basecbs 17088  Hom chom 17149   Func cfunc 17745   Faith cfth 17795  SetCatcsetc 17966  ExtStrCatcestrc 18014
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-rmo 3352  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-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-wun 10643  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-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-func 17749  df-fth 17797  df-setc 17967  df-estrc 18015
This theorem is referenced by:  equivestrcsetc  18045
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