Theorem estrcval 16764

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
estrcval.h	⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
estrcval.o	⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))

Assertion

Ref	Expression
estrcval	⊢ (𝜑 → 𝐶 = {⟨(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.

Step	Hyp	Ref	Expression
1		estrcval.c	. 2 ⊢ 𝐶 = (ExtStrCat‘𝑈)
2		df-estrc 16763	. . . 4 ⊢ ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩})
3	2	a1i 11	. . 3 ⊢ (𝜑 → ExtStrCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩}))
4		simpr 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
5	4	opeq2d 4409	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
6		eqidd 2623	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
7	4, 4, 6	mpt2eq123dv 6717	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
8		estrcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
9	8	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝐻 = (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
10	7, 9	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = 𝐻)
11	10	opeq2d 4409	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
12	4	sqxpeqd 5141	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
13		eqidd 2623	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)) = (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))
14	12, 4, 13	mpt2eq123dv 6717	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
15		estrcval.o	. . . . . . 7 ⊢ (𝜑 → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
16	15	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧 ∈ 𝑈 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))))
17	14, 16	eqtr4d 2659	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓))) = · )
18	17	opeq2d 4409	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
19	5, 11, 18	tpeq123d 4283	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑢, 𝑦 ∈ 𝑢 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧 ∈ 𝑢 ↦ (𝑔 ∈ ((Base‘𝑧) ↑𝑚 (Base‘(2nd ‘𝑣))), 𝑓 ∈ ((Base‘(2nd ‘𝑣)) ↑𝑚 (Base‘(1st ‘𝑣))) ↦ (𝑔 ∘ 𝑓)))⟩} = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
20		estrcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
21		elex 3212	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝑈 ∈ V)
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
23		tpex 6957	. . . 4 ⊢ {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
24	23	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
25	3, 19, 22, 24	fvmptd 6288	. 2 ⊢ (𝜑 → (ExtStrCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
26	1, 25	syl5eq 2668	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {ctp 4181  ⟨cop 4183   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857  ndxcnx 15854  Basecbs 15857  Hom chom 15952  compcco 15953  ExtStrCatcestrc 16762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-estrc 16763

This theorem is referenced by:  estrcbas  16765  estrchomfval  16766  estrccofval  16769  dfrngc2  41972  dfringc2  42018