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Theorem rescval2 16535
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)
4 xpexg 7002 . . . . 5 ((𝑆𝑊𝑆𝑊) → (𝑆 × 𝑆) ∈ V)
53, 3, 4syl2anc 694 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ V)
6 fnex 6522 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
72, 5, 6syl2anc 694 . . 3 (𝜑𝐻 ∈ V)
8 rescval.1 . . . 4 𝐷 = (𝐶cat 𝐻)
98rescval 16534 . . 3 ((𝐶𝑉𝐻 ∈ V) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
101, 7, 9syl2anc 694 . 2 (𝜑𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11 fndm 6028 . . . . . . 7 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
122, 11syl 17 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1312dmeqd 5358 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
14 dmxpid 5377 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1513, 14syl6eq 2701 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
1615oveq2d 6706 . . 3 (𝜑 → (𝐶s dom dom 𝐻) = (𝐶s 𝑆))
1716oveq1d 6705 . 2 (𝜑 → ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
1810, 17eqtrd 2685 1 (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  cop 4216   × cxp 5141  dom cdm 5143   Fn wfn 5921  cfv 5926  (class class class)co 6690  ndxcnx 15901   sSet csts 15902  s cress 15905  Hom chom 15999  cat cresc 16515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-resc 16518
This theorem is referenced by:  rescbas  16536  reschom  16537  rescco  16539  rescabs  16540  rescabs2  16541  dfrngc2  42297  dfringc2  42343  rngcresringcat  42355  rngcrescrhm  42410  rngcrescrhmALTV  42428
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