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

Theorem rescval2 16259
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 6835 . . . . 5 ((𝑆𝑊𝑆𝑊) → (𝑆 × 𝑆) ∈ V)
53, 3, 4syl2anc 690 . . . 4 (𝜑 → (𝑆 × 𝑆) ∈ V)
6 fnex 6363 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
72, 5, 6syl2anc 690 . . 3 (𝜑𝐻 ∈ V)
8 rescval.1 . . . 4 𝐷 = (𝐶cat 𝐻)
98rescval 16258 . . 3 ((𝐶𝑉𝐻 ∈ V) → 𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
101, 7, 9syl2anc 690 . 2 (𝜑𝐷 = ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11 fndm 5889 . . . . . . 7 (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
122, 11syl 17 . . . . . 6 (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
1312dmeqd 5234 . . . . 5 (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
14 dmxpid 5252 . . . . 5 dom (𝑆 × 𝑆) = 𝑆
1513, 14syl6eq 2659 . . . 4 (𝜑 → dom dom 𝐻 = 𝑆)
1615oveq2d 6542 . . 3 (𝜑 → (𝐶s dom dom 𝐻) = (𝐶s 𝑆))
1716oveq1d 6541 . 2 (𝜑 → ((𝐶s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
1810, 17eqtrd 2643 1 (𝜑𝐷 = ((𝐶s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1976  Vcvv 3172  cop 4130   × cxp 5025  dom cdm 5027   Fn wfn 5784  cfv 5789  (class class class)co 6526  ndxcnx 15640   sSet csts 15641  s cress 15644  Hom chom 15727  cat cresc 16239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-resc 16242
This theorem is referenced by:  rescbas  16260  reschom  16261  rescco  16263  rescabs  16264  rescabs2  16265  dfrngc2  41745  dfringc2  41791  rngcresringcat  41803  rngcrescrhm  41858  rngcrescrhmALTV  41877
  Copyright terms: Public domain W3C validator