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| Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcnidlem | Structured version Visualization version GIF version | ||
| Description: Lemma for prstcnid 49128 and prstchomval 49136. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
| 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) |
| Ref | Expression |
|---|---|
| prstcnidlem | ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | prstcnid.c | . . . 4 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) | |
| 2 | prstcnid.k | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Proset ) | |
| 3 | 1, 2 | prstcval 49126 | . . 3 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
| 4 | 3 | fveq2d 6908 | . 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉))) |
| 5 | prstcnid.e | . . 3 ⊢ 𝐸 = Slot (𝐸‘ndx) | |
| 6 | prstcnid.no | . . 3 ⊢ (𝐸‘ndx) ≠ (comp‘ndx) | |
| 7 | 5, 6 | setsnid 17241 | . 2 ⊢ (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉)) = (𝐸‘((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
| 8 | 4, 7 | eqtr4di 2794 | 1 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∅c0 4332 {csn 4624 〈cop 4630 × cxp 5681 ‘cfv 6559 (class class class)co 7429 1oc1o 8495 sSet csts 17196 Slot cslot 17214 ndxcnx 17226 lecple 17300 Hom chom 17304 compcco 17305 Proset cproset 18334 ProsetToCatcprstc 49124 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-res 5695 df-iota 6512 df-fun 6561 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-sets 17197 df-slot 17215 df-prstc 49125 |
| This theorem is referenced by: prstcnid 49128 prstchomval 49136 |
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