Theorem ishoma 10706

Description: Definition of (hom ‘T).

Hypotheses

Ref	Expression
ishoma.1	⊢ O = dom (id ‘T)
ishoma.2	⊢ M = dom (dom ‘T)
ishoma.3	⊢ D = (dom ‘T)
ishoma.4	⊢ C = (cod ‘T)

Assertion

Ref	Expression
ishoma	⊢ (T ∈ Cat → (hom ‘T) = {⟨⟨a, b⟩, c⟩∣(a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})})

Distinct variable groups:   C,c   D,c   M,c   T,a,b,c,f

Proof of Theorem ishoma

Step	Hyp	Ref	Expression
1		fveq2 3738	. . . . . . 7 ⊢ (x = T → (id ‘x) = (id ‘T))
2	1	dmeqd 3327	. . . . . 6 ⊢ (x = T → dom (id ‘x) = dom (id ‘T))
3		ishoma.1	. . . . . 6 ⊢ O = dom (id ‘T)
4	2, 3	syl6eqr 1532	. . . . 5 ⊢ (x = T → dom (id ‘x) = O)
5	4	eleq2d 1548	. . . 4 ⊢ (x = T → (a ∈ dom (id ‘x) ↔ a ∈ O))
6	4	eleq2d 1548	. . . 4 ⊢ (x = T → (b ∈ dom (id ‘x) ↔ b ∈ O))
7		fveq2 3738	. . . . . . . . . 10 ⊢ (x = T → (dom ‘x) = (dom ‘T))
8	7	dmeqd 3327	. . . . . . . . 9 ⊢ (x = T → dom (dom ‘x) = dom (dom ‘T))
9		ishoma.2	. . . . . . . . 9 ⊢ M = dom (dom ‘T)
10	8, 9	syl6eqr 1532	. . . . . . . 8 ⊢ (x = T → dom (dom ‘x) = M)
11	10	eleq2d 1548	. . . . . . 7 ⊢ (x = T → (f ∈ dom (dom ‘x) ↔ f ∈ M))
12		ishoma.3	. . . . . . . . . 10 ⊢ D = (dom ‘T)
13	7, 12	syl6eqr 1532	. . . . . . . . 9 ⊢ (x = T → (dom ‘x) = D)
14	13	fveq1d 3740	. . . . . . . 8 ⊢ (x = T → ((dom ‘x) ‘f) = (D ‘f))
15	14	eqeq1d 1490	. . . . . . 7 ⊢ (x = T → (((dom ‘x) ‘f) = a ↔ (D ‘f) = a))
16		fveq2 3738	. . . . . . . . . 10 ⊢ (x = T → (cod ‘x) = (cod ‘T))
17		ishoma.4	. . . . . . . . . 10 ⊢ C = (cod ‘T)
18	16, 17	syl6eqr 1532	. . . . . . . . 9 ⊢ (x = T → (cod ‘x) = C)
19	18	fveq1d 3740	. . . . . . . 8 ⊢ (x = T → ((cod ‘x) ‘f) = (C ‘f))
20	19	eqeq1d 1490	. . . . . . 7 ⊢ (x = T → (((cod ‘x) ‘f) = b ↔ (C ‘f) = b))
21	11, 15, 20	3anbi123d 897	. . . . . 6 ⊢ (x = T → ((f ∈ dom (dom ‘x) ⋀ ((dom ‘x) ‘f) = a ⋀ ((cod ‘x) ‘f) = b) ↔ (f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)))
22	21	abbidv 1584	. . . . 5 ⊢ (x = T → {f∣(f ∈ dom (dom ‘x) ⋀ ((dom ‘x) ‘f) = a ⋀ ((cod ‘x) ‘f) = b)} = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})
23	22	eqeq2d 1493	. . . 4 ⊢ (x = T → (c = {f∣(f ∈ dom (dom ‘x) ⋀ ((dom ‘x) ‘f) = a ⋀ ((cod ‘x) ‘f) = b)} ↔ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)}))
24	5, 6, 23	3anbi123d 897	. . 3 ⊢ (x = T → ((a ∈ dom (id ‘x) ⋀ b ∈ dom (id ‘x) ⋀ c = {f∣(f ∈ dom (dom ‘x) ⋀ ((dom ‘x) ‘f) = a ⋀ ((cod ‘x) ‘f) = b)}) ↔ (a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})))
25	24	oprabbidv 4010	. 2 ⊢ (x = T → {⟨⟨a, b⟩, c⟩∣(a ∈ dom (id ‘x) ⋀ b ∈ dom (id ‘x) ⋀ c = {f∣(f ∈ dom (dom ‘x) ⋀ ((dom ‘x) ‘f) = a ⋀ ((cod ‘x) ‘f) = b)})} = {⟨⟨a, b⟩, c⟩∣(a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})})
26		df-hom 10705	. 2 ⊢ hom = {⟨x, y⟩∣(x ∈ Cat ⋀ y = {⟨⟨a, b⟩, c⟩∣(a ∈ dom (id ‘x) ⋀ b ∈ dom (id ‘x) ⋀ c = {f∣(f ∈ dom (dom ‘x) ⋀ ((dom ‘x) ‘f) = a ⋀ ((cod ‘x) ‘f) = b)})})}
27		fvex 3746	. . . 4 ⊢ (id ‘T) ∈ V
28	27	dmex 3374	. . 3 ⊢ dom (id ‘T) ∈ V
29	3	eleq2i 1545	. . . . . 6 ⊢ (a ∈ O ↔ a ∈ dom (id ‘T))
30	3	eleq2i 1545	. . . . . 6 ⊢ (b ∈ O ↔ b ∈ dom (id ‘T))
31		pm4.2 170	. . . . . 6 ⊢ (c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)} ↔ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})
32	29, 30, 31	3anbi123i 826	. . . . 5 ⊢ ((a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)}) ↔ (a ∈ dom (id ‘T) ⋀ b ∈ dom (id ‘T) ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)}))
33		df-3an 781	. . . . 5 ⊢ ((a ∈ dom (id ‘T) ⋀ b ∈ dom (id ‘T) ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)}) ↔ ((a ∈ dom (id ‘T) ⋀ b ∈ dom (id ‘T)) ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)}))
34	32, 33	bitr 173	. . . 4 ⊢ ((a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)}) ↔ ((a ∈ dom (id ‘T) ⋀ b ∈ dom (id ‘T)) ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)}))
35	34	oprabbii 4011	. . 3 ⊢ {⟨⟨a, b⟩, c⟩∣(a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})} = {⟨⟨a, b⟩, c⟩∣((a ∈ dom (id ‘T) ⋀ b ∈ dom (id ‘T)) ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})}
36	28, 28, 35	oprabex2 4035	. 2 ⊢ {⟨⟨a, b⟩, c⟩∣(a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})} ∈ V
37	25, 26, 36	fvopab4 3794	1 ⊢ (T ∈ Cat → (hom ‘T) = {⟨⟨a, b⟩, c⟩∣(a ∈ O ⋀ b ∈ O ⋀ c = {f∣(f ∈ M ⋀ (D ‘f) = a ⋀ (C ‘f) = b)})})

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 779   = wceq 960   ∈ wcel 962  {cab 1470  dom cdm 3184   ‘cfv 3196  {copab2 3978  domcdom_ 10635  codccod_ 10636  idcid_ 10637  Catccat 10676  homchom 10704

This theorem is referenced by:  ishomb 10707

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-rep 2706  ax-sep 2716  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 781  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-rex 1657  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-fv 3212  df-oprab 3980  df-hom 10705

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