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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 16720 | . 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 45962 | . 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
8 | 3, 7 | eqtr4id 2790 | 1 ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 {csn 4527 〈cop 4533 × cxp 5534 ‘cfv 6358 (class class class)co 7191 1oc1o 8173 ndxcnx 16663 sSet csts 16664 Slot cslot 16665 lecple 16756 Hom chom 16760 compcco 16761 Proset cproset 17754 ProsetToCatcprstc 45959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-res 5548 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-slot 16670 df-sets 16673 df-prstc 45960 |
This theorem is referenced by: prstcbas 45964 prstcleval 45965 prstcocval 45967 |
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