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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstchom | Structured version Visualization version GIF version |
Description: Hom-sets of the
constructed category are dependent on the preorder.
Note that prstchom.x and prstchom.y are redundant here due to our definition of ProsetToCat. However, this should not be assumed as it is definition-dependent. Therefore, the two hypotheses are added for explicitness. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
prstcnid.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
prstcnid.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
prstchom.l | ⊢ (𝜑 → ≤ = (le‘𝐶)) |
prstchom.e | ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) |
prstchom.x | ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐶)) |
prstchom.y | ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐶)) |
Ref | Expression |
---|---|
prstchom | ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prstchom.e | . . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) | |
2 | prstcnid.c | . . . 4 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
3 | prstcnid.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
4 | prstchom.l | . . . 4 ⊢ (𝜑 → ≤ = (le‘𝐶)) | |
5 | 2, 3, 4 | prstchomval 46307 | . . 3 ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶)) |
6 | 1, 5 | eqtr4d 2782 | . 2 ⊢ (𝜑 → 𝐻 = ( ≤ × {1o})) |
7 | 1oex 8294 | . . 3 ⊢ 1o ∈ V | |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → 1o ∈ V) |
9 | 1n0 8300 | . . 3 ⊢ 1o ≠ ∅ | |
10 | 9 | a1i 11 | . 2 ⊢ (𝜑 → 1o ≠ ∅) |
11 | 6, 8, 10 | fvconstrn0 46136 | 1 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 Vcvv 3430 ∅c0 4261 {csn 4566 class class class wbr 5078 × cxp 5586 ‘cfv 6430 (class class class)co 7268 1oc1o 8274 Basecbs 16893 lecple 16950 Hom chom 16954 Proset cproset 17992 ProsetToCatcprstc 46295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-en 8708 df-dom 8709 df-sdom 8710 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ple 16963 df-hom 16967 df-cco 16968 df-prstc 46296 |
This theorem is referenced by: prstchom2 46311 postc 46315 |
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