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Mirrors > Home > MPE Home > Th. List > Mathboxes > thincepi | Structured version Visualization version GIF version |
Description: In a thin category, all morphisms are epimorphisms. The converse does not hold. See grptcepi 45992. (Contributed by Zhi Wang, 24-Sep-2024.) |
Ref | Expression |
---|---|
thincid.c | ⊢ (𝜑 → 𝐶 ∈ ThinCat) |
thincid.b | ⊢ 𝐵 = (Base‘𝐶) |
thincid.h | ⊢ 𝐻 = (Hom ‘𝐶) |
thincid.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
thincmon.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
thincepi.e | ⊢ 𝐸 = (Epi‘𝐶) |
Ref | Expression |
---|---|
thincepi | ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | thincmon.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
2 | 1 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑌𝐻𝑧) ∧ ℎ ∈ (𝑌𝐻𝑧))) → 𝑌 ∈ 𝐵) |
3 | simpr1 1196 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑌𝐻𝑧) ∧ ℎ ∈ (𝑌𝐻𝑧))) → 𝑧 ∈ 𝐵) | |
4 | simpr2 1197 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑌𝐻𝑧) ∧ ℎ ∈ (𝑌𝐻𝑧))) → 𝑔 ∈ (𝑌𝐻𝑧)) | |
5 | simpr3 1198 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑌𝐻𝑧) ∧ ℎ ∈ (𝑌𝐻𝑧))) → ℎ ∈ (𝑌𝐻𝑧)) | |
6 | thincid.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
7 | thincid.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
8 | thincid.c | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ThinCat) | |
9 | 8 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑌𝐻𝑧) ∧ ℎ ∈ (𝑌𝐻𝑧))) → 𝐶 ∈ ThinCat) |
10 | 2, 3, 4, 5, 6, 7, 9 | thincmo2 45925 | . . . . 5 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑌𝐻𝑧) ∧ ℎ ∈ (𝑌𝐻𝑧))) → 𝑔 = ℎ) |
11 | 10 | a1d 25 | . . . 4 ⊢ ((𝜑 ∧ (𝑧 ∈ 𝐵 ∧ 𝑔 ∈ (𝑌𝐻𝑧) ∧ ℎ ∈ (𝑌𝐻𝑧))) → ((𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑧)𝑓) = (ℎ(〈𝑋, 𝑌〉(comp‘𝐶)𝑧)𝑓) → 𝑔 = ℎ)) |
12 | 11 | ralrimivvva 3103 | . . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑌𝐻𝑧)∀ℎ ∈ (𝑌𝐻𝑧)((𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑧)𝑓) = (ℎ(〈𝑋, 𝑌〉(comp‘𝐶)𝑧)𝑓) → 𝑔 = ℎ)) |
13 | eqid 2736 | . . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
14 | thincepi.e | . . . 4 ⊢ 𝐸 = (Epi‘𝐶) | |
15 | 8 | thinccd 45922 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) |
16 | thincid.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
17 | 6, 7, 13, 14, 15, 16, 1 | isepi2 17200 | . . 3 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐸𝑌) ↔ (𝑓 ∈ (𝑋𝐻𝑌) ∧ ∀𝑧 ∈ 𝐵 ∀𝑔 ∈ (𝑌𝐻𝑧)∀ℎ ∈ (𝑌𝐻𝑧)((𝑔(〈𝑋, 𝑌〉(comp‘𝐶)𝑧)𝑓) = (ℎ(〈𝑋, 𝑌〉(comp‘𝐶)𝑧)𝑓) → 𝑔 = ℎ)))) |
18 | 12, 17 | mpbiran2d 708 | . 2 ⊢ (𝜑 → (𝑓 ∈ (𝑋𝐸𝑌) ↔ 𝑓 ∈ (𝑋𝐻𝑌))) |
19 | 18 | eqrdv 2734 | 1 ⊢ (𝜑 → (𝑋𝐸𝑌) = (𝑋𝐻𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 〈cop 4533 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 compcco 16761 Epicepi 17188 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-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 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-hom 16773 df-cco 16774 df-cat 17125 df-cid 17126 df-oppc 17169 df-mon 17189 df-epi 17190 df-thinc 45917 |
This theorem is referenced by: (None) |
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