Theorem catcval 18002

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 18001	. . 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 3440	. . . . . 6 ⊢ 𝑢 ∈ V
4	3	inex1 5250	. . . . 5 ⊢ (𝑢 ∩ Cat) ∈ V
5	4	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Cat) ∈ V)
6		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈)
7	6	ineq1d 4164	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Cat) = (𝑈 ∩ Cat))
8		catcval.b	. . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
9	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → 𝐵 = (𝑈 ∩ Cat))
10	7, 9	eqtr4d 2769	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (𝑢 ∩ Cat) = 𝐵)
11		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
12	11	opeq2d 4827	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Base‘ndx), 𝑏⟩ = ⟨(Base‘ndx), 𝐵⟩)
13		eqidd 2732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 Func 𝑦) = (𝑥 Func 𝑦))
14	11, 11, 13	mpoeq123dv 7416	. . . . . . 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
17	14, 16	eqtr4d 2769	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦)) = 𝐻)
18	17	opeq2d 4827	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → ⟨(Hom ‘ndx), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 Func 𝑦))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
19	11	sqxpeqd 5643	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 = 𝑈) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
20		eqidd 2732	. . . . . . . 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 7416	. . . . . . 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 726	. . . . . . 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 2769	. . . . . 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 4827	. . . . 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 4696	. . . 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 3882	. . 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 3460	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
30		tpex 7674	. . . 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 6932	. 2 ⊢ (𝜑 → (CatCat‘𝑈) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
33	1, 32	eqtrid 2778	1 ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⦋csb 3845   ∩ cin 3896  {ctp 4575  ⟨cop 4577   × cxp 5609  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  2^nd c2nd 7915  ndxcnx 17099  Basecbs 17115  Hom chom 17167  compcco 17168  Catccat 17565   Func cfunc 17756   ∘_func ccofu 17758  CatCatccatc 18000

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-tp 4576  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-iota 6432  df-fun 6478  df-fv 6484  df-oprab 7345  df-mpo 7346  df-catc 18001

This theorem is referenced by:  catcbas  18003  catchomfval  18004  catccofval  18006