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Theorem estrcval 18179
Description: Value of the category of extensible structures (in a universe). (Contributed by AV, 7-Mar-2020.)
Hypotheses
Ref Expression
estrcval.c 𝐶 = (ExtStrCat‘𝑈)
estrcval.u (𝜑𝑈𝑉)
estrcval.h (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
estrcval.o (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
Assertion
Ref Expression
estrcval (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Distinct variable groups:   𝑓,𝑔,𝑣,𝑥,𝑦,𝑧   𝜑,𝑣,𝑥,𝑦,𝑧   𝑣,𝑈,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   · (𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑈(𝑓,𝑔)   𝐻(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)

Proof of Theorem estrcval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 estrcval.c . 2 𝐶 = (ExtStrCat‘𝑈)
2 df-estrc 18178 . . 3 ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩})
3 simpr 484 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝑢 = 𝑈)
43opeq2d 4885 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
5 eqidd 2736 . . . . . . 7 ((𝜑𝑢 = 𝑈) → ((Base‘𝑦) ↑m (Base‘𝑥)) = ((Base‘𝑦) ↑m (Base‘𝑥)))
63, 3, 5mpoeq123dv 7508 . . . . . 6 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
7 estrcval.h . . . . . . 7 (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
87adantr 480 . . . . . 6 ((𝜑𝑢 = 𝑈) → 𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
96, 8eqtr4d 2778 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))) = 𝐻)
109opeq2d 4885 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
113sqxpeqd 5721 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
12 eqidd 2736 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)) = (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))
1311, 3, 12mpoeq123dv 7508 . . . . . 6 ((𝜑𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
14 estrcval.o . . . . . . 7 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1514adantr 480 . . . . . 6 ((𝜑𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))))
1613, 15eqtr4d 2778 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓))) = · )
1716opeq2d 4885 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
184, 10, 17tpeq123d 4753 . . 3 ((𝜑𝑢 = 𝑈) → {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ ((Base‘𝑦) ↑m (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑m (Base‘(2nd𝑣))), 𝑓 ∈ ((Base‘(2nd𝑣)) ↑m (Base‘(1st𝑣))) ↦ (𝑔𝑓)))⟩} = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
19 estrcval.u . . . 4 (𝜑𝑈𝑉)
2019elexd 3502 . . 3 (𝜑𝑈 ∈ V)
21 tpex 7765 . . . 4 {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
2221a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
232, 18, 20, 22fvmptd2 7024 . 2 (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
241, 23eqtrid 2787 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {ctp 4635  cop 4637   × cxp 5687  ccom 5693  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  m cmap 8865  ndxcnx 17227  Basecbs 17245  Hom chom 17309  compcco 17310  ExtStrCatcestrc 18177
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-sep 5302  ax-nul 5312  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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  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-iota 6516  df-fun 6565  df-fv 6571  df-oprab 7435  df-mpo 7436  df-estrc 18178
This theorem is referenced by:  estrcbas  18180  estrchomfval  18181  estrccofval  18184  dfrngc2  20645  dfringc2  20674
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