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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 17245 | . 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 49154 | . 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
| 8 | 3, 7 | eqtr4id 2796 | 1 ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {csn 4626 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 sSet csts 17200 Slot cslot 17218 ndxcnx 17230 lecple 17304 Hom chom 17308 compcco 17309 Proset cproset 18338 ProsetToCatcprstc 49151 |
| 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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-sets 17201 df-slot 17219 df-prstc 49152 |
| This theorem is referenced by: prstcbas 49156 prstcleval 49157 prstclevalOLD 49158 prstcocval 49160 prstcocvalOLD 49161 |
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