Theorem cmdfval 50702

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 490	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
2		simpl 488	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
3	1, 2	oveq12d 7430	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑐) = (𝐷 Func 𝐶))
4	1, 2	oveq12d 7430	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑐) = (𝐷 FuncCat 𝐶))
5	2, 4	oveq12d 7430	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐 UP (𝑑 FuncCat 𝑐)) = (𝐶 UP (𝐷 FuncCat 𝐶)))
6		oveq12 7421	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐Δfunc𝑑) = (𝐶Δfunc𝐷))
7		eqidd 2762	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑓 = 𝑓)
8	5, 6, 7	oveq123d 7433	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓) = ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))
9	3, 8	mpteq12dv 5192	. . 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 50698	. . 3 ⊢ Colimit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((𝑐Δfunc𝑑)(𝑐 UP (𝑑 FuncCat 𝑐))𝑓)))
11		ovex 7445	. . . 4 ⊢ (𝐷 Func 𝐶) ∈ V
12	11	mptex 7221	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) ∈ V
13	9, 10, 12	ovmpoa 7567	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)))
14		reldmcmd 50700	. . . 4 ⊢ Rel dom Colimit
15	14	ovprc 7450	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = ∅)
16		ancom 466	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐷 ∈ V ∧ 𝐶 ∈ V))
17		reldmfunc 50127	. . . . . . 7 ⊢ Rel dom Func
18	17	ovprc 7450	. . . . . 6 ⊢ (¬ (𝐷 ∈ V ∧ 𝐶 ∈ V) → (𝐷 Func 𝐶) = ∅)
19	16, 18	sylnbi 333	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐷 Func 𝐶) = ∅)
20	19	mpteq1d 5195	. . . 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 6673	. . . 4 ⊢ (𝑓 ∈ ∅ ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) = ∅
22	20, 21	eqtrdi 2812	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)) = ∅)
23	15, 22	eqtr4d 2799	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓)))
24	13, 23	pm2.61i 184	1 ⊢ (𝐶 Colimit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((𝐶Δfunc𝐷)(𝐶 UP (𝐷 FuncCat 𝐶))𝑓))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279   ↦ cmpt 5186  (class class class)co 7412   Func cfunc 18009   FuncCat cfuc 18100  Δ_funccdiag 18366   UP cup 50225   Colimit ccmd 50696

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 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-func 18013  df-cmd 50698

This theorem is used by:  cmdrcl  50704  reldmcmd2  50706  cmdfval2  50708  cmdpropd  50710