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Theorem rescval 17099
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 3498 . . 3 (𝐶𝑉𝐶 ∈ V)
3 elex 3498 . . 3 (𝐻𝑊𝐻 ∈ V)
4 simpl 486 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
5 simpr 488 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
65dmeqd 5762 . . . . . . 7 ((𝑐 = 𝐶 = 𝐻) → dom = dom 𝐻)
76dmeqd 5762 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → dom dom = dom dom 𝐻)
84, 7oveq12d 7169 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → (𝑐s dom dom ) = (𝐶s dom dom 𝐻))
95opeq2d 4796 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
108, 9oveq12d 7169 . . . 4 ((𝑐 = 𝐶 = 𝐻) → ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11 df-resc 17083 . . . 4 cat = (𝑐 ∈ V, ∈ V ↦ ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩))
12 ovex 7184 . . . 4 ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
1310, 11, 12ovmpoa 7300 . . 3 ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
142, 3, 13syl2an 598 . 2 ((𝐶𝑉𝐻𝑊) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
151, 14syl5eq 2871 1 ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cop 4556  dom cdm 5543  cfv 6345  (class class class)co 7151  ndxcnx 16482   sSet csts 16483  s cress 16486  Hom chom 16578  cat cresc 17080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6304  df-fun 6347  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-resc 17083
This theorem is referenced by:  rescval2  17100
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