Theorem catcval 17356

Description: Value of the category of categories (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)

Hypotheses

Ref	Expression
catcval.c	⊢ 𝐶 = (CatCat‘𝑈)
catcval.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
catcval.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
catcval.h	⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
catcval.o	⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))

Assertion

Ref	Expression
catcval	⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Distinct variable groups:   𝑥,𝑣,𝑦,𝑧,𝐵   𝜑,𝑣,𝑥,𝑦,𝑧   𝑣,𝑈,𝑥,𝑦,𝑧   𝑓,𝑔,𝑣,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐵(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 catcval

Dummy variables 𝑢 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catcval.c	. 2 ⊢ 𝐶 = (CatCat‘𝑈)
2		df-catc 17355	. . 3 ⊢ CatCat = (𝑢 ∈ V ↦ ⦋(𝑢 ∩ Cat) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩})
3		vex 3497	. . . . . 6 ⊢ 𝑢 ∈ V
4	3	inex1 5221	. . . . 5 ⊢ (𝑢 ∩ Cat) ∈ V
5	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Cat) ∈ V)
6		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
7	6	ineq1d 4188	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Cat) = (𝑈 ∩ Cat))
8		catcval.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
9	8	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝐵 = (𝑈 ∩ Cat))
10	7, 9	eqtr4d 2859	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Cat) = 𝐵)
11		simpr 487	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
12	11	opeq2d 4810	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
13		eqidd 2822	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 Func 𝑦) = (𝑥 Func 𝑦))
14	11, 11, 13	mpoeq123dv 7229	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
15		catcval.h	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
16	15	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
17	14, 16	eqtr4d 2859	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦)) = 𝐻)
18	17	opeq2d 4810	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
19	11	sqxpeqd 5587	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
20		eqidd 2822	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)) = (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))
21	19, 11, 20	mpoeq123dv 7229	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
22		catcval.o	. . . . . . . 8 ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
23	22	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))))
24	21, 23	eqtr4d 2859	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓))) = · )
25	24	opeq2d 4810	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
26	12, 18, 25	tpeq123d 4684	. . . 4 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → {⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
27	5, 10, 26	csbied2 3920	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ⦋(𝑢 ∩ Cat) / 𝑏⦌{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) Func 𝑧), 𝑓 ∈ ( Func ‘𝑣) ↦ (𝑔 ∘func 𝑓)))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
28		catcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
29	28	elexd 3514	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
30		tpex 7470	. . . 4 ⊢ {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
31	30	a1i 11	. . 3 ⊢ (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
32	2, 27, 29, 31	fvmptd2 6776	. 2 ⊢ (𝜑 → (CatCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
33	1, 32	syl5eq 2868	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⦋csb 3883   ∩ cin 3935  {ctp 4571  ⟨cop 4573   × cxp 5553  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  2^nd c2nd 7688  ndxcnx 16480  Basecbs 16483  Hom chom 16576  compcco 16577  Catccat 16935   Func cfunc 17124   ∘_func ccofu 17126  CatCatccatc 17354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fv 6363  df-oprab 7160  df-mpo 7161  df-catc 17355

This theorem is referenced by:  catcbas  17357  catchomfval  17358  catccofval  17360