Theorem homffval 17784

Description: Value of the functionalized Hom-set operation. (Contributed by Mario Carneiro, 4-Jan-2017.) (Proof shortened by AV, 1-Mar-2024.)

Hypotheses

Ref	Expression
homffval.f	⊢ 𝐹 = (Homf ‘𝐶)
homffval.b	⊢ 𝐵 = (Base‘𝐶)
homffval.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
homffval	⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐻,𝑦

Allowed substitution hints:   𝐹(𝑥, 𝑦)

Proof of Theorem homffval

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		homffval.f	. 2 ⊢ 𝐹 = (Homf ‘𝐶)
2		fveq2 6882	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = (Base‘𝐶))
3		homffval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4	2, 3	eqtr4di 2815	. . . . 5 ⊢ (𝑐 = 𝐶 → (Base‘𝑐) = 𝐵)
5		fveq2 6882	. . . . . . 7 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = (Hom ‘𝐶))
6		homffval.h	. . . . . . 7 ⊢ 𝐻 = (Hom ‘𝐶)
7	5, 6	eqtr4di 2815	. . . . . 6 ⊢ (𝑐 = 𝐶 → (Hom ‘𝑐) = 𝐻)
8	7	oveqd 7434	. . . . 5 ⊢ (𝑐 = 𝐶 → (𝑥(Hom ‘𝑐)𝑦) = (𝑥𝐻𝑦))
9	4, 4, 8	mpoeq123dv 7492	. . . 4 ⊢ (𝑐 = 𝐶 → (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
10		df-homf 17764	. . . 4 ⊢ Homf = (𝑐 ∈ V ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ (𝑥(Hom ‘𝑐)𝑦)))
11	3	fvexi 6896	. . . . 5 ⊢ 𝐵 ∈ V
12	11, 11	mpoex 8082	. . . 4 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) ∈ V
13	9, 10, 12	fvmpt 6990	. . 3 ⊢ (𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
14		fvprc 6874	. . . 4 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅)
15		fvprc 6874	. . . . . . 7 ⊢ (¬ 𝐶 ∈ V → (Base‘𝐶) = ∅)
16	3, 15	eqtrid 2809	. . . . . 6 ⊢ (¬ 𝐶 ∈ V → 𝐵 = ∅)
17	16	olcd 888	. . . . 5 ⊢ (¬ 𝐶 ∈ V → (𝐵 = ∅ ∨ 𝐵 = ∅))
18		0mpo0 7500	. . . . 5 ⊢ ((𝐵 = ∅ ∨ 𝐵 = ∅) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
19	17, 18	syl 18	. . . 4 ⊢ (¬ 𝐶 ∈ V → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)) = ∅)
20	14, 19	eqtr4d 2800	. . 3 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦)))
21	13, 20	pm2.61i 184	. 2 ⊢ (Homf ‘𝐶) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))
22	1, 21	eqtri 2785	1 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥𝐻𝑦))

This proof depends on syntax axioms:  ¬ wn 3   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∅c0 4282  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  Basecbs 17307  Hom chom 17359  Hom_f chomf 17760

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-homf 17764

This theorem is used by:  fnhomeqhomf  17785  homfval  17786  homffn  17787  homfeq  17788  oppchomf  17814  reschomf  17926  homf0  49943