Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prstcnidlem Structured version   Visualization version   GIF version

Theorem prstcnidlem 48786
Description: Lemma for prstcnid 48787 and prstchomval 48795. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
Hypotheses
Ref Expression
prstcnid.c (𝜑𝐶 = (ProsetToCat‘𝐾))
prstcnid.k (𝜑𝐾 ∈ Proset )
prstcnid.e 𝐸 = Slot (𝐸‘ndx)
prstcnid.no (𝐸‘ndx) ≠ (comp‘ndx)
Assertion
Ref Expression
prstcnidlem (𝜑 → (𝐸𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))

Proof of Theorem prstcnidlem
StepHypRef Expression
1 prstcnid.c . . . 4 (𝜑𝐶 = (ProsetToCat‘𝐾))
2 prstcnid.k . . . 4 (𝜑𝐾 ∈ Proset )
31, 2prstcval 48785 . . 3 (𝜑𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
43fveq2d 6905 . 2 (𝜑 → (𝐸𝐶) = (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)
75, 6setsnid 17232 . 2 (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)) = (𝐸‘((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
84, 7eqtr4di 2791 1 (𝜑 → (𝐸𝐶) = (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
  Copyright terms: Public domain W3C validator