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Theorem rescval2 17876
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 7770 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ V)
5 fnex 7237 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
62, 4, 5syl2anc 584 . . 3 (𝜑𝐻 ∈ V)
7 rescval.1 . . . 4 𝐷 = (𝐶cat 𝐻)
87rescval 17875 . . 3 ((𝐶𝑉𝐻 ∈ V) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
91, 6, 8syl2anc 584 . 2 (𝜑𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
102fndmd 6674 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1110dmeqd 5919 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12 dmxpid 5944 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1311, 12eqtrdi 2791 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
1413oveq2d 7447 . . 3 (𝜑 → (𝐶s dom dom 𝐻) = (𝐶s 𝑆))
1514oveq1d 7446 . 2 (𝜑 → ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
169, 15eqtrd 2775 1 (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  cop 4637   × cxp 5687  dom cdm 5689   Fn wfn 6558  cfv 6563  (class class class)co 7431   sSet csts 17197  ndxcnx 17227  s cress 17274  Hom chom 17309  cat cresc 17856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-resc 17859
This theorem is referenced by:  rescbas  17877  rescbasOLD  17878  reschom  17879  rescco  17881  resccoOLD  17882  rescabs  17883  rescabsOLD  17884  rescabs2  17885  dfrngc2  20645  dfringc2  20674  rngcresringcat  20686  rngcrescrhm  20701  rngcrescrhmALTV  48124
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