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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 16635 | . 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 45789 | . 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
8 | 3, 7 | eqtr4id 2792 | 1 ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 {csn 4513 〈cop 4519 × cxp 5517 ‘cfv 6333 (class class class)co 7164 1oc1o 8117 ndxcnx 16576 sSet csts 16577 Slot cslot 16578 lecple 16668 Hom chom 16672 compcco 16673 Proset cproset 17645 ProsetToCatcprstc 45786 |
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 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-res 5531 df-iota 6291 df-fun 6335 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-slot 16583 df-sets 16586 df-prstc 45787 |
This theorem is referenced by: prstcbas 45791 prstcleval 45792 prstcocval 45794 |
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