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Mirrors > Home > MPE Home > Th. List > Mathboxes > subthinc | Structured version Visualization version GIF version |
Description: A subcategory of a thin category is thin. (Contributed by Zhi Wang, 30-Sep-2024.) |
Ref | Expression |
---|---|
subthinc.1 | ⊢ 𝐷 = (𝐶 ↾cat 𝐽) |
subthinc.j | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subthinc.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
Ref | Expression |
---|---|
subthinc | ⊢ (𝜑 → 𝐷 ∈ ThinCat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subthinc.1 | . . 3 ⊢ 𝐷 = (𝐶 ↾cat 𝐽) | |
2 | eqid 2736 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
3 | subthinc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
4 | subthinc.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
5 | eqidd 2737 | . . . 4 ⊢ (𝜑 → dom dom 𝐽 = dom dom 𝐽) | |
6 | 4, 5 | subcfn 17301 | . . 3 ⊢ (𝜑 → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
7 | 4, 6, 2 | subcss1 17302 | . . 3 ⊢ (𝜑 → dom dom 𝐽 ⊆ (Base‘𝐶)) |
8 | 1, 2, 3, 6, 7 | rescbas 17288 | . 2 ⊢ (𝜑 → dom dom 𝐽 = (Base‘𝐷)) |
9 | 1, 2, 3, 6, 7 | reschom 17289 | . 2 ⊢ (𝜑 → 𝐽 = (Hom ‘𝐷)) |
10 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝐽 ∈ (Subcat‘𝐶)) |
11 | 6 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝐽 Fn (dom dom 𝐽 × dom dom 𝐽)) |
12 | eqid 2736 | . . . . 5 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
13 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑥 ∈ dom dom 𝐽) | |
14 | simprr 773 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑦 ∈ dom dom 𝐽) | |
15 | 10, 11, 12, 13, 14 | subcss2 17303 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → (𝑥𝐽𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦)) |
16 | 3 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝐶 ∈ ThinCat) |
17 | 7 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → dom dom 𝐽 ⊆ (Base‘𝐶)) |
18 | 17, 13 | sseldd 3888 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑥 ∈ (Base‘𝐶)) |
19 | 17, 14 | sseldd 3888 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → 𝑦 ∈ (Base‘𝐶)) |
20 | 16, 18, 19, 2, 12 | thincmo 45926 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)) |
21 | mosssn2 45778 | . . . . 5 ⊢ (∃*𝑓 𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦) ↔ ∃𝑓(𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓}) | |
22 | 20, 21 | sylib 221 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃𝑓(𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓}) |
23 | sstr2 3894 | . . . . 5 ⊢ ((𝑥𝐽𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦) → ((𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓} → (𝑥𝐽𝑦) ⊆ {𝑓})) | |
24 | 23 | eximdv 1925 | . . . 4 ⊢ ((𝑥𝐽𝑦) ⊆ (𝑥(Hom ‘𝐶)𝑦) → (∃𝑓(𝑥(Hom ‘𝐶)𝑦) ⊆ {𝑓} → ∃𝑓(𝑥𝐽𝑦) ⊆ {𝑓})) |
25 | 15, 22, 24 | sylc 65 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃𝑓(𝑥𝐽𝑦) ⊆ {𝑓}) |
26 | mosssn2 45778 | . . 3 ⊢ (∃*𝑓 𝑓 ∈ (𝑥𝐽𝑦) ↔ ∃𝑓(𝑥𝐽𝑦) ⊆ {𝑓}) | |
27 | 25, 26 | sylibr 237 | . 2 ⊢ ((𝜑 ∧ (𝑥 ∈ dom dom 𝐽 ∧ 𝑦 ∈ dom dom 𝐽)) → ∃*𝑓 𝑓 ∈ (𝑥𝐽𝑦)) |
28 | 1, 4 | subccat 17308 | . 2 ⊢ (𝜑 → 𝐷 ∈ Cat) |
29 | 8, 9, 27, 28 | isthincd 45934 | 1 ⊢ (𝜑 → 𝐷 ∈ ThinCat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ∃*wmo 2537 ⊆ wss 3853 {csn 4527 × cxp 5534 dom cdm 5536 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 ↾cat cresc 17267 Subcatcsubc 17268 ThinCatcthinc 45916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-hom 16773 df-cco 16774 df-cat 17125 df-cid 17126 df-homf 17127 df-ssc 17269 df-resc 17270 df-subc 17271 df-thinc 45917 |
This theorem is referenced by: (None) |
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