Theorem lmdfval 50583

Description: Function value of Limit. (Contributed by Zhi Wang, 14-Nov-2025.)

Assertion

Ref	Expression
lmdfval	⊢ (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))

Distinct variable groups:   𝐶,𝑓   𝐷,𝑓

Proof of Theorem lmdfval

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 7435	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑐) = (𝐷 Func 𝐶))
4	2	fveq2d 6886	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (oppCat‘𝑐) = (oppCat‘𝐶))
5	1, 2	oveq12d 7435	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑐) = (𝐷 FuncCat 𝐶))
6	5	fveq2d 6886	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (oppCat‘(𝑑 FuncCat 𝑐)) = (oppCat‘(𝐷 FuncCat 𝐶)))
7	4, 6	oveq12d 7435	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐))) = ((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶))))
8		oveq12 7426	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐Δfunc𝑑) = (𝐶Δfunc𝐷))
9	8	fveq2d 6886	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ( oppFunc ‘(𝑐Δfunc𝑑)) = ( oppFunc ‘(𝐶Δfunc𝐷)))
10		eqidd 2763	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑓 = 𝑓)
11	7, 9, 10	oveq123d 7438	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓) = (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))
12	3, 11	mpteq12dv 5196	. . 3 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)))
13		df-lmd 50579	. . 3 ⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
14		ovex 7450	. . . 4 ⊢ (𝐷 Func 𝐶) ∈ V
15	14	mptex 7226	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) ∈ V
16	12, 13, 15	ovmpoa 7572	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)))
17		reldmlmd 50581	. . . 4 ⊢ Rel dom Limit
18	17	ovprc 7455	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Limit 𝐷) = ∅)
19		ancom 466	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐷 ∈ V ∧ 𝐶 ∈ V))
20		reldmfunc 50009	. . . . . . 7 ⊢ Rel dom Func
21	20	ovprc 7455	. . . . . 6 ⊢ (¬ (𝐷 ∈ V ∧ 𝐶 ∈ V) → (𝐷 Func 𝐶) = ∅)
22	19, 21	sylnbi 333	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐷 Func 𝐶) = ∅)
23	22	mpteq1d 5199	. . . 4 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) = (𝑓 ∈ ∅ ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)))
24		mpt0 6678	. . . 4 ⊢ (𝑓 ∈ ∅ ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) = ∅
25	23, 24	eqtrdi 2813	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) = ∅)
26	18, 25	eqtr4d 2800	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)))
27	16, 26	pm2.61i 184	1 ⊢ (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3453  ∅c0 4282   ↦ cmpt 5190  ‘cfv 6537  (class class class)co 7417  oppCatcoppc 17805   Func cfunc 17949   FuncCat cfuc 18040  Δ_funccdiag 18306   oppFunc coppf 50056   UP cup 50107   Limit clmd 50577

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-pr 5402

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-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-func 17953  df-lmd 50579

This theorem is used by:  lmdrcl  50585  reldmlmd2  50587  lmdfval2  50589  lmdpropd  50591