Theorem cmdfval 50119

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 7385	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑐) = (𝐷 Func 𝐶))
4	1, 2	oveq12d 7385	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑐) = (𝐷 FuncCat 𝐶))
5	2, 4	oveq12d 7385	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐 UP (𝑑 FuncCat 𝑐)) = (𝐶 UP (𝐷 FuncCat 𝐶)))
6		oveq12 7376	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐Δfunc𝑑) = (𝐶Δfunc𝐷))
7		eqidd 2738	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑓 = 𝑓)
8	5, 6, 7	oveq123d 7388	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))
9	3, 8	mpteq12dv 5173	. . 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 50115	. . 3 ⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
11		ovex 7400	. . . 4 ⊢ (𝐷 Func 𝐶) ∈ V
12	11	mptex 7178	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) ∈ V
13	9, 10, 12	ovmpoa 7522	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)))
14		reldmcmd 50117	. . . 4 ⊢ Rel dom Colimit
15	14	ovprc 7405	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = ∅)
16		ancom 460	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐷 ∈ V ∧ 𝐶 ∈ V))
17		reldmfunc 49544	. . . . . . 7 ⊢ Rel dom Func
18	17	ovprc 7405	. . . . . 6 ⊢ (¬ (𝐷 ∈ V ∧ 𝐶 ∈ V) → (𝐷 Func 𝐶) = ∅)
19	16, 18	sylnbi 330	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐷 Func 𝐶) = ∅)
20	19	mpteq1d 5176	. . . 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 6641	. . . 4 ⊢ (𝑓 ∈ ∅ ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) = ∅
22	20, 21	eqtrdi 2788	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) = ∅)
23	15, 22	eqtr4d 2775	. 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 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274   ↦ cmpt 5167  (class class class)co 7367   Func cfunc 17821   FuncCat cfuc 17912  Δ_funccdiag 18178   UP cup 49642   Colimit ccmd 50113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6455  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7370  df-oprab 7371  df-mpo 7372  df-func 17825  df-cmd 50115

This theorem is referenced by:  cmdrcl  50121  reldmcmd2  50123  cmdfval2  50125  cmdpropd  50127