Theorem cmdfval 49645

Description: Function value of Colimit. (Contributed by Zhi Wang, 12-Nov-2025.)

Assertion

Ref	Expression
cmdfval	⊢ (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))

Distinct variable groups:   𝐶,𝑓   𝐷,𝑓

Proof of Theorem cmdfval

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
2		simpl 482	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
3	1, 2	oveq12d 7367	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑐) = (𝐷 Func 𝐶))
4	1, 2	oveq12d 7367	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑐) = (𝐷 FuncCat 𝐶))
5	2, 4	oveq12d 7367	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐 UP (𝑑 FuncCat 𝑐)) = (𝐶 UP (𝐷 FuncCat 𝐶)))
6		oveq12 7358	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐Δfunc𝑑) = (𝐶Δfunc𝐷))
7		eqidd 2730	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑓 = 𝑓)
8	5, 6, 7	oveq123d 7370	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))
9	3, 8	mpteq12dv 5179	. . 3 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)))
10		df-cmd 49641	. . 3 ⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
11		ovex 7382	. . . 4 ⊢ (𝐷 Func 𝐶) ∈ V
12	11	mptex 7159	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) ∈ V
13	9, 10, 12	ovmpoa 7504	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)))
14		reldmcmd 49643	. . . 4 ⊢ Rel dom Colimit
15	14	ovprc 7387	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = ∅)
16		ancom 460	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐷 ∈ V ∧ 𝐶 ∈ V))
17		reldmfunc 49070	. . . . . . 7 ⊢ Rel dom Func
18	17	ovprc 7387	. . . . . 6 ⊢ (¬ (𝐷 ∈ V ∧ 𝐶 ∈ V) → (𝐷 Func 𝐶) = ∅)
19	16, 18	sylnbi 330	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐷 Func 𝐶) = ∅)
20	19	mpteq1d 5182	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) = (𝑓 ∈ ∅ ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)))
21		mpt0 6624	. . . 4 ⊢ (𝑓 ∈ ∅ ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) = ∅
22	20, 21	eqtrdi 2780	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) = ∅)
23	15, 22	eqtr4d 2767	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)))
24	13, 23	pm2.61i 182	1 ⊢ (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3436  ∅c0 4284   ↦ cmpt 5173  (class class class)co 7349   Func cfunc 17761   FuncCat cfuc 17852  Δ_funccdiag 18118   UP cup 49168   Colimit ccmd 49639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-func 17765  df-cmd 49641

This theorem is referenced by:  cmdrcl  49647  reldmcmd2  49649  cmdfval2  49651  cmdpropd  49653