Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rescval2 Structured version   Visualization version   GIF version

Theorem rescval2 17077
 Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
rescval.1 𝐷 = (𝐶cat 𝐻)
rescval2.1 (𝜑𝐶𝑉)
rescval2.2 (𝜑𝑆𝑊)
rescval2.3 (𝜑𝐻 Fn (𝑆 × 𝑆))
Assertion
Ref Expression
rescval2 (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Proof of Theorem rescval2
StepHypRef Expression
1 rescval2.1 . . 3 (𝜑𝐶𝑉)
2 rescval2.3 . . . 4 (𝜑𝐻 Fn (𝑆 × 𝑆))
3 rescval2.2 . . . . 5 (𝜑𝑆𝑊)
43, 3xpexd 7452 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ V)
5 fnex 6956 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
62, 4, 5syl2anc 586 . . 3 (𝜑𝐻 ∈ V)
7 rescval.1 . . . 4 𝐷 = (𝐶cat 𝐻)
87rescval 17076 . . 3 ((𝐶𝑉𝐻 ∈ V) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
91, 6, 8syl2anc 586 . 2 (𝜑𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
10 fndm 6431 . . . . . . 7 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
112, 10syl 17 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1211dmeqd 5750 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
13 dmxpid 5776 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1412, 13syl6eq 2871 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
1514oveq2d 7149 . . 3 (𝜑 → (𝐶s dom dom 𝐻) = (𝐶s 𝑆))
1615oveq1d 7148 . 2 (𝜑 → ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
179, 16eqtrd 2855 1 (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3473  ⟨cop 4549   × cxp 5529  dom cdm 5531   Fn wfn 6326  ‘cfv 6331  (class class class)co 7133  ndxcnx 16459   sSet csts 16460   ↾s cress 16463  Hom chom 16555   ↾cat cresc 17057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-resc 17060 This theorem is referenced by:  rescbas  17078  reschom  17079  rescco  17081  rescabs  17082  rescabs2  17083  dfrngc2  44388  dfringc2  44434  rngcresringcat  44446  rngcrescrhm  44501  rngcrescrhmALTV  44519
 Copyright terms: Public domain W3C validator