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Theorem rescval 16883
Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypothesis
Ref Expression
rescval.1 𝐷 = (𝐶cat 𝐻)
Assertion
Ref Expression
rescval ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Proof of Theorem rescval
Dummy variables 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rescval.1 . 2 𝐷 = (𝐶cat 𝐻)
2 elex 3414 . . 3 (𝐶𝑉𝐶 ∈ V)
3 elex 3414 . . 3 (𝐻𝑊𝐻 ∈ V)
4 simpl 476 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
5 simpr 479 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
65dmeqd 5573 . . . . . . 7 ((𝑐 = 𝐶 = 𝐻) → dom = dom 𝐻)
76dmeqd 5573 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → dom dom = dom dom 𝐻)
84, 7oveq12d 6942 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → (𝑐s dom dom ) = (𝐶s dom dom 𝐻))
95opeq2d 4645 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
108, 9oveq12d 6942 . . . 4 ((𝑐 = 𝐶 = 𝐻) → ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11 df-resc 16867 . . . 4 cat = (𝑐 ∈ V, ∈ V ↦ ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩))
12 ovex 6956 . . . 4 ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
1310, 11, 12ovmpt2a 7070 . . 3 ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
142, 3, 13syl2an 589 . 2 ((𝐶𝑉𝐻𝑊) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
151, 14syl5eq 2826 1 ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  Vcvv 3398  cop 4404  dom cdm 5357  cfv 6137  (class class class)co 6924  ndxcnx 16263   sSet csts 16264  s cress 16267  Hom chom 16360  cat cresc 16864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-resc 16867
This theorem is referenced by:  rescval2  16884
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