![]() |
Mathbox for Zhi Wang |
< Previous
Next >
Nearby theorems |
|
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 47848. See prstchom2 47863 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 7445 | . . . 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 47859 | . . . . . . 7 ⊢ (𝜑 → ( ≤ × {1o}) = (Hom ‘𝐶)) |
7 | 2, 6 | eqtr4d 2774 | . . . . . 6 ⊢ (𝜑 → 𝐻 = ( ≤ × {1o})) |
8 | 1oex 8482 | . . . . . . 7 ⊢ 1o ∈ V | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1o ∈ V) |
10 | 1n0 8494 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 1o ≠ ∅) |
12 | 7, 9, 11 | fvconstr 47687 | . . . . 5 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) = 1o)) |
13 | 12 | biimpa 476 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → (𝑋𝐻𝑌) = 1o) |
14 | eqeng 8988 | . . . 4 ⊢ ((𝑋𝐻𝑌) ∈ V → ((𝑋𝐻𝑌) = 1o → (𝑋𝐻𝑌) ≈ 1o)) | |
15 | 1, 13, 14 | mpsyl 68 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → (𝑋𝐻𝑌) ≈ 1o) |
16 | euen1b 9033 | . . 3 ⊢ ((𝑋𝐻𝑌) ≈ 1o ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
17 | 15, 16 | sylib 217 | . 2 ⊢ ((𝜑 ∧ 𝑋 ≤ 𝑌) → ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) |
18 | euex 2570 | . . . 4 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
19 | n0 4346 | . . . 4 ⊢ ((𝑋𝐻𝑌) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝑋𝐻𝑌)) | |
20 | 18, 19 | sylibr 233 | . . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌) → (𝑋𝐻𝑌) ≠ ∅) |
21 | 7, 9, 11 | fvconstrn0 47688 | . . . 4 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ (𝑋𝐻𝑌) ≠ ∅)) |
22 | 21 | biimpar 477 | . . 3 ⊢ ((𝜑 ∧ (𝑋𝐻𝑌) ≠ ∅) → 𝑋 ≤ 𝑌) |
23 | 20, 22 | sylan2 592 | . 2 ⊢ ((𝜑 ∧ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌)) → 𝑋 ≤ 𝑌) |
24 | 17, 23 | impbida 798 | 1 ⊢ (𝜑 → (𝑋 ≤ 𝑌 ↔ ∃!𝑓 𝑓 ∈ (𝑋𝐻𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∃!weu 2561 ≠ wne 2939 Vcvv 3473 ∅c0 4322 {csn 4628 class class class wbr 5148 × cxp 5674 ‘cfv 6543 (class class class)co 7412 1oc1o 8465 ≈ cen 8942 lecple 17211 Hom chom 17215 Proset cproset 18256 ProsetToCatcprstc 47847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ple 17224 df-hom 17228 df-cco 17229 df-prstc 47848 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |