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

Theorem rescval2 16877
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 7240 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ V)
5 fnex 6755 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
62, 4, 5syl2anc 579 . . 3 (𝜑𝐻 ∈ V)
7 rescval.1 . . . 4 𝐷 = (𝐶cat 𝐻)
87rescval 16876 . . 3 ((𝐶𝑉𝐻 ∈ V) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
91, 6, 8syl2anc 579 . 2 (𝜑𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
10 fndm 6237 . . . . . . 7 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
112, 10syl 17 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1211dmeqd 5573 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
13 dmxpid 5592 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1412, 13syl6eq 2830 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
1514oveq2d 6940 . . 3 (𝜑 → (𝐶s dom dom 𝐻) = (𝐶s 𝑆))
1615oveq1d 6939 . 2 (𝜑 → ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
179, 16eqtrd 2814 1 (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1601  wcel 2107  Vcvv 3398  cop 4404   × cxp 5355  dom cdm 5357   Fn wfn 6132  cfv 6137  (class class class)co 6924  ndxcnx 16256   sSet csts 16257  s cress 16260  Hom chom 16353  cat cresc 16857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-resc 16860
This theorem is referenced by:  rescbas  16878  reschom  16879  rescco  16881  rescabs  16882  rescabs2  16883  dfrngc2  42997  dfringc2  43043  rngcresringcat  43055  rngcrescrhm  43110  rngcrescrhmALTV  43128
  Copyright terms: Public domain W3C validator