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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcnidlem | Structured version Visualization version GIF version |
Description: Lemma for prstcnid 48787 and prstchomval 48795. (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 48785 | . . 3 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
4 | 3 | fveq2d 6905 | . 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 17232 | . 2 ⊢ (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉)) = (𝐸‘((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
8 | 4, 7 | eqtr4di 2791 | 1 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 ∅c0 4339 {csn 4630 〈cop 4636 × cxp 5681 ‘cfv 6558 (class class class)co 7425 1oc1o 8492 sSet csts 17186 Slot cslot 17204 ndxcnx 17216 lecple 17294 Hom chom 17298 compcco 17299 Proset cproset 18339 ProsetToCatcprstc 48783 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-res 5695 df-iota 6510 df-fun 6560 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-sets 17187 df-slot 17205 df-prstc 48784 |
This theorem is referenced by: prstcnid 48787 prstchomval 48795 |
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