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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 17169 | . 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 47994 | . 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
8 | 3, 7 | eqtr4id 2786 | 1 ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 {csn 4624 〈cop 4630 × cxp 5670 ‘cfv 6542 (class class class)co 7414 1oc1o 8473 sSet csts 17123 Slot cslot 17141 ndxcnx 17153 lecple 17231 Hom chom 17235 compcco 17236 Proset cproset 18276 ProsetToCatcprstc 47991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-res 5684 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-sets 17124 df-slot 17142 df-prstc 47992 |
This theorem is referenced by: prstcbas 47996 prstcleval 47997 prstclevalOLD 47998 prstcocval 48000 prstcocvalOLD 48001 |
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