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Theorem rescval 17810
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 3490 . . 3 (𝐶𝑉𝐶 ∈ V)
3 elex 3490 . . 3 (𝐻𝑊𝐻 ∈ V)
4 simpl 482 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
5 simpr 484 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
65dmeqd 5908 . . . . . . 7 ((𝑐 = 𝐶 = 𝐻) → dom = dom 𝐻)
76dmeqd 5908 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → dom dom = dom dom 𝐻)
84, 7oveq12d 7438 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → (𝑐s dom dom ) = (𝐶s dom dom 𝐻))
95opeq2d 4881 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
108, 9oveq12d 7438 . . . 4 ((𝑐 = 𝐶 = 𝐻) → ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11 df-resc 17794 . . . 4 cat = (𝑐 ∈ V, ∈ V ↦ ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩))
12 ovex 7453 . . . 4 ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
1310, 11, 12ovmpoa 7576 . . 3 ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
142, 3, 13syl2an 595 . 2 ((𝐶𝑉𝐻𝑊) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
151, 14eqtrid 2780 1 ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3471  cop 4635  dom cdm 5678  cfv 6548  (class class class)co 7420   sSet csts 17132  ndxcnx 17162  s cress 17209  Hom chom 17244  cat cresc 17791
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-resc 17794
This theorem is referenced by:  rescval2  17811
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