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Theorem rescval 16256
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 3184 . . 3 (𝐶𝑉𝐶 ∈ V)
3 elex 3184 . . 3 (𝐻𝑊𝐻 ∈ V)
4 simpl 471 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → 𝑐 = 𝐶)
5 simpr 475 . . . . . . . 8 ((𝑐 = 𝐶 = 𝐻) → = 𝐻)
65dmeqd 5235 . . . . . . 7 ((𝑐 = 𝐶 = 𝐻) → dom = dom 𝐻)
76dmeqd 5235 . . . . . 6 ((𝑐 = 𝐶 = 𝐻) → dom dom = dom dom 𝐻)
84, 7oveq12d 6545 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → (𝑐s dom dom ) = (𝐶s dom dom 𝐻))
95opeq2d 4341 . . . . 5 ((𝑐 = 𝐶 = 𝐻) → ⟨(Hom ‘ndx), ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
108, 9oveq12d 6545 . . . 4 ((𝑐 = 𝐶 = 𝐻) → ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11 df-resc 16240 . . . 4 cat = (𝑐 ∈ V, ∈ V ↦ ((𝑐s dom dom ) sSet ⟨(Hom ‘ndx), ⟩))
12 ovex 6555 . . . 4 ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
1310, 11, 12ovmpt2a 6667 . . 3 ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
142, 3, 13syl2an 492 . 2 ((𝐶𝑉𝐻𝑊) → (𝐶cat 𝐻) = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
151, 14syl5eq 2655 1 ((𝐶𝑉𝐻𝑊) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130  dom cdm 5028  cfv 5790  (class class class)co 6527  ndxcnx 15638   sSet csts 15639  s cress 15642  Hom chom 15725  cat cresc 16237
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-iota 5754  df-fun 5792  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-resc 16240
This theorem is referenced by:  rescval2  16257
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