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Theorem rescval 16708
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 3352 . . 3 (𝐶𝑉𝐶 ∈ V)
3 elex 3352 . . 3 (𝐻𝑊𝐻 ∈ V)
4 simpl 474 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
5 simpr 479 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
65dmeqd 5481 . . . . . . 7 ((𝑐 = 𝐶 = 𝐻) → dom = dom 𝐻)
76dmeqd 5481 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → dom dom = dom dom 𝐻)
84, 7oveq12d 6832 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → (𝑐s dom dom ) = (𝐶s dom dom 𝐻))
95opeq2d 4560 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
108, 9oveq12d 6832 . . . 4 ((𝑐 = 𝐶 = 𝐻) → ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11 df-resc 16692 . . . 4 cat = (𝑐 ∈ V, ∈ V ↦ ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩))
12 ovex 6842 . . . 4 ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
1310, 11, 12ovmpt2a 6957 . . 3 ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
142, 3, 13syl2an 495 . 2 ((𝐶𝑉𝐻𝑊) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
151, 14syl5eq 2806 1 ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cop 4327  dom cdm 5266  cfv 6049  (class class class)co 6814  ndxcnx 16076   sSet csts 16077  s cress 16080  Hom chom 16174  cat cresc 16689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-resc 16692
This theorem is referenced by:  rescval2  16709
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