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Theorem rescval2 17814
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 7754 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ V)
5 fnex 7229 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
62, 4, 5syl2anc 582 . . 3 (𝜑𝐻 ∈ V)
7 rescval.1 . . . 4 𝐷 = (𝐶cat 𝐻)
87rescval 17813 . . 3 ((𝐶𝑉𝐻 ∈ V) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
91, 6, 8syl2anc 582 . 2 (𝜑𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
102fndmd 6660 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1110dmeqd 5908 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12 dmxpid 5932 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1311, 12eqtrdi 2781 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
1413oveq2d 7435 . . 3 (𝜑 → (𝐶s dom dom 𝐻) = (𝐶s 𝑆))
1514oveq1d 7434 . 2 (𝜑 → ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
169, 15eqtrd 2765 1 (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  cop 4636   × cxp 5676  dom cdm 5678   Fn wfn 6544  cfv 6549  (class class class)co 7419   sSet csts 17135  ndxcnx 17165  s cress 17212  Hom chom 17247  cat cresc 17794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-resc 17797
This theorem is referenced by:  rescbas  17815  rescbasOLD  17816  reschom  17817  rescco  17819  resccoOLD  17820  rescabs  17821  rescabsOLD  17822  rescabs2  17823  dfrngc2  20573  dfringc2  20602  rngcresringcat  20614  rngcrescrhm  20629  rngcrescrhmALTV  47528
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