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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcnidlem | Structured version Visualization version GIF version |
Description: Lemma for prstcnid 46347 and prstchomval 46355. (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 46345 | . . 3 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
4 | 3 | fveq2d 6778 | . 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 16910 | . 2 ⊢ (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉)) = (𝐸‘((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
8 | 4, 7 | eqtr4di 2796 | 1 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∅c0 4256 {csn 4561 〈cop 4567 × cxp 5587 ‘cfv 6433 (class class class)co 7275 1oc1o 8290 sSet csts 16864 Slot cslot 16882 ndxcnx 16894 lecple 16969 Hom chom 16973 compcco 16974 Proset cproset 18011 ProsetToCatcprstc 46343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-res 5601 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-sets 16865 df-slot 16883 df-prstc 46344 |
This theorem is referenced by: prstcnid 46347 prstchomval 46355 |
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