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| Mirrors > Home > MPE Home > Th. List > Mathboxes > thinccisod | Structured version Visualization version GIF version | ||
| Description: Two thin categories are isomorphic if the induced preorders are order-isomorphic (deduction form). Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 22-Sep-2025.) |
| Ref | Expression |
|---|---|
| thinccisod.c | ⊢ 𝐶 = (CatCat‘𝑈) |
| thinccisod.r | ⊢ 𝑅 = (Base‘𝑋) |
| thinccisod.s | ⊢ 𝑆 = (Base‘𝑌) |
| thinccisod.h | ⊢ 𝐻 = (Hom ‘𝑋) |
| thinccisod.j | ⊢ 𝐽 = (Hom ‘𝑌) |
| thinccisod.u | ⊢ (𝜑 → 𝑈 ∈ 𝑉) |
| thinccisod.x | ⊢ (𝜑 → 𝑋 ∈ 𝑈) |
| thinccisod.y | ⊢ (𝜑 → 𝑌 ∈ 𝑈) |
| thinccisod.xt | ⊢ (𝜑 → 𝑋 ∈ ThinCat) |
| thinccisod.yt | ⊢ (𝜑 → 𝑌 ∈ ThinCat) |
| thinccisod.f | ⊢ (𝜑 → 𝐹:𝑅–1-1-onto→𝑆) |
| thinccisod.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) |
| Ref | Expression |
|---|---|
| thinccisod | ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | thinccisod.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑅–1-1-onto→𝑆) | |
| 2 | f1of 6846 | . . . . 5 ⊢ (𝐹:𝑅–1-1-onto→𝑆 → 𝐹:𝑅⟶𝑆) | |
| 3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝑅⟶𝑆) |
| 4 | thinccisod.r | . . . . 5 ⊢ 𝑅 = (Base‘𝑋) | |
| 5 | fvexd 6919 | . . . . 5 ⊢ (𝜑 → (Base‘𝑋) ∈ V) | |
| 6 | 4, 5 | eqeltrid 2844 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ V) |
| 7 | 3, 6 | fexd 7245 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
| 8 | thinccisod.1 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) | |
| 9 | 8 | ralrimivva 3201 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) |
| 10 | 9, 1 | jca 511 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) ∧ 𝐹:𝑅–1-1-onto→𝑆)) |
| 11 | fveq1 6903 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
| 12 | fveq1 6903 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦)) | |
| 13 | 11, 12 | oveq12d 7447 | . . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ((𝐹‘𝑥)𝐽(𝐹‘𝑦))) |
| 14 | 13 | eqeq1d 2738 | . . . . . 6 ⊢ (𝑓 = 𝐹 → (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) |
| 15 | 14 | bibi2d 342 | . . . . 5 ⊢ (𝑓 = 𝐹 → (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ↔ ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))) |
| 16 | 15 | 2ralbidv 3220 | . . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ↔ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))) |
| 17 | f1oeq1 6834 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝑅–1-1-onto→𝑆 ↔ 𝐹:𝑅–1-1-onto→𝑆)) | |
| 18 | 16, 17 | anbi12d 632 | . . 3 ⊢ (𝑓 = 𝐹 → ((∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆) ↔ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) ∧ 𝐹:𝑅–1-1-onto→𝑆))) |
| 19 | 7, 10, 18 | spcedv 3597 | . 2 ⊢ (𝜑 → ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) |
| 20 | thinccisod.c | . . 3 ⊢ 𝐶 = (CatCat‘𝑈) | |
| 21 | eqid 2736 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 22 | thinccisod.s | . . 3 ⊢ 𝑆 = (Base‘𝑌) | |
| 23 | thinccisod.h | . . 3 ⊢ 𝐻 = (Hom ‘𝑋) | |
| 24 | thinccisod.j | . . 3 ⊢ 𝐽 = (Hom ‘𝑌) | |
| 25 | thinccisod.u | . . 3 ⊢ (𝜑 → 𝑈 ∈ 𝑉) | |
| 26 | thinccisod.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑈) | |
| 27 | thinccisod.xt | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ThinCat) | |
| 28 | 27 | thinccd 49049 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ Cat) |
| 29 | 26, 28 | elind 4199 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑈 ∩ Cat)) |
| 30 | 20, 21, 25 | catcbas 18142 | . . . 4 ⊢ (𝜑 → (Base‘𝐶) = (𝑈 ∩ Cat)) |
| 31 | 29, 30 | eleqtrrd 2843 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
| 32 | thinccisod.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑈) | |
| 33 | thinccisod.yt | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ThinCat) | |
| 34 | 33 | thinccd 49049 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ Cat) |
| 35 | 32, 34 | elind 4199 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑈 ∩ Cat)) |
| 36 | 35, 30 | eleqtrrd 2843 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
| 37 | 20, 21, 4, 22, 23, 24, 25, 31, 36, 27, 33 | thincciso 49078 | . 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆))) |
| 38 | 19, 37 | mpbird 257 | 1 ⊢ (𝜑 → 𝑋( ≃𝑐 ‘𝐶)𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3060 Vcvv 3479 ∩ cin 3949 ∅c0 4332 class class class wbr 5141 ⟶wf 6555 –1-1-onto→wf1o 6558 ‘cfv 6559 (class class class)co 7429 Basecbs 17243 Hom chom 17304 Catccat 17703 ≃𝑐 ccic 17835 CatCatccatc 18139 ThinCatcthinc 49043 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-1st 8010 df-2nd 8011 df-supp 8182 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-nn 12263 df-2 12325 df-3 12326 df-4 12327 df-5 12328 df-6 12329 df-7 12330 df-8 12331 df-9 12332 df-n0 12523 df-z 12610 df-dec 12730 df-uz 12875 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17244 df-hom 17317 df-cco 17318 df-cat 17707 df-cid 17708 df-sect 17787 df-inv 17788 df-iso 17789 df-cic 17836 df-func 17899 df-idfu 17900 df-cofu 17901 df-full 17947 df-fth 17948 df-catc 18140 df-thinc 49044 |
| This theorem is referenced by: oduoppcciso 49146 |
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