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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcnid | Structured version Visualization version GIF version |
Description: Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
prstcnid.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
prstcnid.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
prstcnid.e | ⊢ 𝐸 = Slot (𝐸‘ndx) |
prstcnid.no | ⊢ (𝐸‘ndx) ≠ (comp‘ndx) |
prstcnid.nh | ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx) |
Ref | Expression |
---|---|
prstcnid | ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prstcnid.e | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
2 | prstcnid.nh | . . 3 ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx) | |
3 | 1, 2 | setsnid 17243 | . 2 ⊢ (𝐸‘𝐾) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉)) |
4 | prstcnid.c | . . 3 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
5 | prstcnid.k | . . 3 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
6 | prstcnid.no | . . 3 ⊢ (𝐸‘ndx) ≠ (comp‘ndx) | |
7 | 4, 5, 1, 6 | prstcnidlem 48866 | . 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
8 | 3, 7 | eqtr4id 2794 | 1 ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 {csn 4631 〈cop 4637 × cxp 5687 ‘cfv 6563 (class class class)co 7431 1oc1o 8498 sSet csts 17197 Slot cslot 17215 ndxcnx 17227 lecple 17305 Hom chom 17309 compcco 17310 Proset cproset 18350 ProsetToCatcprstc 48863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-res 5701 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17198 df-slot 17216 df-prstc 48864 |
This theorem is referenced by: prstcbas 48868 prstcleval 48869 prstclevalOLD 48870 prstcocval 48872 prstcocvalOLD 48873 |
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