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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstchom2ALT | Structured version Visualization version GIF version |
Description: Hom-sets of the constructed category are dependent on the preorder. This proof depends on the definition df-prstc 45850. See prstchom2 45863 for a version that does not depend on the definition. (Contributed by Zhi Wang, 20-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prstcnid.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
prstcnid.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
prstchom.l | ⊢ (𝜑 → ≤ = (le‘𝐶)) |
prstchom.e | ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) |
Ref | Expression |
---|---|
prstchom2ALT | ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7215 | . . . 4 ⊢ (𝑋𝐻𝑌) ∈ V | |
2 | prstchom.e | . . . . . . 7 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐶)) | |
3 | prstcnid.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
4 | prstcnid.k | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
5 | prstchom.l | . . . . . . . 8 ⊢ (𝜑 → ≤ = (le‘𝐶)) | |
6 | 3, 4, 5 | prstchomval 45859 | . . . . . . 7 ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶)) |
7 | 2, 6 | eqtr4d 2777 | . . . . . 6 ⊢ (𝜑 → 𝐻 = ( ≤ × {1o})) |
8 | 1oex 8156 | . . . . . . 7 ⊢ 1o ∈ V | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1o ∈ V) |
10 | 1n0 8162 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1o ≠ ∅) |
12 | 7, 9, 11 | fvconstr 45753 | . . . . 5 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) = 1o)) |
13 | 12 | biimpa 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → (𝑋𝐻𝑌) = 1o) |
14 | eqeng 8601 | . . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ((𝑋𝐻𝑌) = 1o → (𝑋𝐻𝑌) ≈ 1o)) | |
15 | 1, 13, 14 | mpsyl 68 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → (𝑋𝐻𝑌) ≈ 1o) |
16 | euen1b 8639 | . . 3 ⊢ ((𝑋𝐻𝑌) ≈ 1o ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
17 | 15, 16 | sylib 221 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
18 | euex 2579 | . . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
19 | n0 4245 | . . . 4 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
20 | 18, 19 | sylibr 237 | . . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅) |
21 | 7, 9, 11 | fvconstrn0 45754 | . . . 4 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) |
22 | 21 | biimpar 481 | . . 3 ⊢ ((𝜑 ∧ (𝑋𝐻𝑌) ≠ ∅) → 𝑋 ≤ 𝑌) |
23 | 20, 22 | sylan2 596 | . 2 ⊢ ((𝜑 ∧ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ≤ 𝑌) |
24 | 17, 23 | impbida 801 | 1 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2114 ∃!weu 2570 ≠ wne 2935 Vcvv 3400 ∅c0 4221 {csn 4526 class class class wbr 5040 × cxp 5533 ‘cfv 6349 (class class class)co 7182 1oc1o 8136 ≈ cen 8564 lecple 16687 Hom chom 16691 Proset cproset 17664 ProsetToCatcprstc 45849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-ndx 16601 df-slot 16602 df-base 16604 df-sets 16605 df-ple 16700 df-hom 16704 df-cco 16705 df-prstc 45850 |
This theorem is referenced by: (None) |
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