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Mirrors > Home > MPE Home > Th. List > rescfth | Structured version Visualization version GIF version |
Description: The inclusion functor from a subcategory is a faithful functor. (Contributed by Mario Carneiro, 27-Jan-2017.) |
Ref | Expression |
---|---|
rescfth.d | ⊢ 𝐷 = (𝐶 ↾cat 𝐽) |
rescfth.i | ⊢ 𝐼 = (idfunc‘𝐷) |
Ref | Expression |
---|---|
rescfth | ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 ∈ (𝐷 Faith 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescfth.d | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐽) | |
2 | 1 | oveq2i 7442 | . . 3 ⊢ (𝐷 Faith 𝐷) = (𝐷 Faith (𝐶 ↾cat 𝐽)) |
3 | fthres2 17986 | . . 3 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (𝐷 Faith (𝐶 ↾cat 𝐽)) ⊆ (𝐷 Faith 𝐶)) | |
4 | 2, 3 | eqsstrid 4044 | . 2 ⊢ (𝐽 ∈ (Subcat‘𝐶) → (𝐷 Faith 𝐷) ⊆ (𝐷 Faith 𝐶)) |
5 | id 22 | . . . . 5 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐽 ∈ (Subcat‘𝐶)) | |
6 | 1, 5 | subccat 17899 | . . . 4 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐷 ∈ Cat) |
7 | rescfth.i | . . . . 5 ⊢ 𝐼 = (idfunc‘𝐷) | |
8 | 7 | idffth 17987 | . . . 4 ⊢ (𝐷 ∈ Cat → 𝐼 ∈ ((𝐷 Full 𝐷) ∩ (𝐷 Faith 𝐷))) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 ∈ ((𝐷 Full 𝐷) ∩ (𝐷 Faith 𝐷))) |
10 | 9 | elin2d 4215 | . 2 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 ∈ (𝐷 Faith 𝐷)) |
11 | 4, 10 | sseldd 3996 | 1 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐼 ∈ (𝐷 Faith 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ‘cfv 6563 (class class class)co 7431 Catccat 17709 ↾cat cresc 17856 Subcatcsubc 17857 idfunccidfu 17906 Full cful 17956 Faith cfth 17957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-hom 17322 df-cco 17323 df-cat 17713 df-cid 17714 df-homf 17715 df-ssc 17858 df-resc 17859 df-subc 17860 df-func 17909 df-idfu 17910 df-full 17958 df-fth 17959 |
This theorem is referenced by: inclfusubc 17995 |
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