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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcnidlem | Structured version Visualization version GIF version |
Description: Lemma for prstcnid 45853 and prstchomval 45859. (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 45851 | . . 3 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
4 | 3 | fveq2d 6690 | . 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 16654 | . 2 ⊢ (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉)) = (𝐸‘((𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉) sSet 〈(comp‘ndx), ∅〉)) |
8 | 4, 7 | eqtr4di 2792 | 1 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet 〈(Hom ‘ndx), ((le‘𝐾) × {1o})〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∅c0 4221 {csn 4526 〈cop 4532 × cxp 5533 ‘cfv 6349 (class class class)co 7182 1oc1o 8136 ndxcnx 16595 sSet csts 16596 Slot cslot 16597 lecple 16687 Hom chom 16691 compcco 16692 Proset cproset 17664 ProsetToCatcprstc 45849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-res 5547 df-iota 6307 df-fun 6351 df-fv 6357 df-ov 7185 df-oprab 7186 df-mpo 7187 df-slot 16602 df-sets 16605 df-prstc 45850 |
This theorem is referenced by: prstcnid 45853 prstchomval 45859 |
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