Theorem catchomfval 18282

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

Hypotheses

Ref	Expression
catcbas.c	⊢ 𝐶 = (CatCat‘𝑈)
catcbas.b	⊢ 𝐵 = (Base‘𝐶)
catcbas.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
catchomfval.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
catchomfval	⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝜑,𝑥,𝑦   𝑥,𝑈,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐻(𝑥, 𝑦)   𝑉(𝑥, 𝑦)

Proof of Theorem catchomfval

Dummy variables 𝑣 𝑧 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catchomfval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
2		catcbas.c	. . . . 5 ⊢ 𝐶 = (CatCat‘𝑈)
3		catcbas.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		catcbas.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
5	2, 4, 3	catcbas 18281	. . . . 5 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Cat))
6		eqidd 2761	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))
7		eqidd 2761	. . . . 5 ⊢ (𝜑 → (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))))
8	2, 3, 5, 6, 7	catcval 18280	. . . 4 ⊢ (𝜑 → 𝐶 = {⟨(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)))⟩})
9	8	fveq2d 6889	. . 3 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘{⟨(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)))⟩}))
10	1, 9	eqtrid 2807	. 2 ⊢ (𝜑 → 𝐻 = (Hom ‘{⟨(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)))⟩}))
11	4	fvexi 6899	. . . 4 ⊢ 𝐵 ∈ V
12	11, 11	mpoex 8092	. . 3 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) ∈ V
13		catstr 18140	. . . 4 ⊢ {⟨(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)))⟩} Struct ⟨1, ;15⟩
14		homid 17588	. . . 4 ⊢ Hom = Slot (Hom ‘ndx)
15		snsstp2 4772	. . . 4 ⊢ {⟨(Hom ‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦))⟩} ⊆ {⟨(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)))⟩}
16	13, 14, 15	strfv 17386	. . 3 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) = (Hom ‘{⟨(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)))⟩}))
17	12, 16	mp1i 14	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)) = (Hom ‘{⟨(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)))⟩}))
18	10, 17	eqtr4d 2798	1 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 Func 𝑦)))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  Vcvv 3450  {ctp 4588  ⟨cop 4834   × cxp 5648  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  2^nd c2nd 8000  1c1 11204  5c5 12403  ;cdc 12817  ndxcnx 17376  Basecbs 17392  Hom chom 17444  compcco 17445   Func cfunc 18034   ∘_func ccofu 18036  CatCatccatc 18278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260  ax-1cn 11261  ax-icn 11262  ax-addcl 11263  ax-addrcl 11264  ax-mulcl 11265  ax-mulrcl 11266  ax-mulcom 11267  ax-addass 11268  ax-mulass 11269  ax-distr 11270  ax-i2m1 11271  ax-1ne0 11272  ax-1rid 11273  ax-rnegex 11274  ax-rrecex 11275  ax-cnre 11276  ax-pre-lttri 11277  ax-pre-lttrn 11278  ax-pre-ltadd 11279  ax-pre-mulgt0 11280

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-er 8720  df-en 8977  df-dom 8978  df-sdom 8979  df-fin 8980  df-pnf 11348  df-mnf 11349  df-xr 11350  df-ltxr 11351  df-le 11352  df-sub 11546  df-neg 11547  df-nn 12339  df-2 12408  df-3 12409  df-4 12410  df-5 12411  df-6 12412  df-7 12413  df-8 12414  df-9 12415  df-n0 12610  df-z 12697  df-dec 12818  df-uz 12969  df-fz 13643  df-struct 17330  df-slot 17365  df-ndx 17377  df-base 17393  df-hom 17457  df-cco 17458  df-catc 18279

This theorem is used by:  catchom  18283  catccofval  18284  catcrcl2  50624