Theorem lmdfval 49471

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 484	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑑 = 𝐷)
2		simpl 482	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑐 = 𝐶)
3	1, 2	oveq12d 7421	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑐) = (𝐷 Func 𝐶))
4	2	fveq2d 6879	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (oppCat‘𝑐) = (oppCat‘𝐶))
5	1, 2	oveq12d 7421	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑐) = (𝐷 FuncCat 𝐶))
6	5	fveq2d 6879	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (oppCat‘(𝑑 FuncCat 𝑐)) = (oppCat‘(𝐷 FuncCat 𝐶)))
7	4, 6	oveq12d 7421	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐))) = ((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶))))
8		oveq12 7412	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐Δfunc𝑑) = (𝐶Δfunc𝐷))
9	8	fveq2d 6879	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (oppFunc‘(𝑐Δfunc𝑑)) = (oppFunc‘(𝐶Δfunc𝐷)))
10		eqidd 2736	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑓 = 𝑓)
11	7, 9, 10	oveq123d 7424	. . . 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 5207	. . 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 49467	. . 3 ⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ ((oppFunc‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
14		ovex 7436	. . . 4 ⊢ (𝐷 Func 𝐶) ∈ V
15	14	mptex 7214	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((oppFunc‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) ∈ V
16	12, 13, 15	ovmpoa 7560	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((oppFunc‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)))
17		reldmlmd 49469	. . . 4 ⊢ Rel dom Limit
18	17	ovprc 7441	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Limit 𝐷) = ∅)
19		ancom 460	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐷 ∈ V ∧ 𝐶 ∈ V))
20		reldmfunc 48990	. . . . . . 7 ⊢ Rel dom Func
21	20	ovprc 7441	. . . . . 6 ⊢ (¬ (𝐷 ∈ V ∧ 𝐶 ∈ V) → (𝐷 Func 𝐶) = ∅)
22	19, 21	sylnbi 330	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐷 Func 𝐶) = ∅)
23	22	mpteq1d 5210	. . . 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 6679	. . . 4 ⊢ (𝑓 ∈ ∅ ↦ ((oppFunc‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) = ∅
25	23, 24	eqtrdi 2786	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((oppFunc‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) = ∅)
26	18, 25	eqtr4d 2773	. 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 182	1 ⊢ (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ ((oppFunc‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓))

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308   ↦ cmpt 5201  ‘cfv 6530  (class class class)co 7403  oppCatcoppc 17721   Func cfunc 17865   FuncCat cfuc 17956  Δ_funccdiag 18222  oppFunccoppf 49019   UP cup 49056   Limit clmd 49465

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-func 17869  df-lmd 49467

This theorem is referenced by:  reldmlmd2  49473  lmdfval2  49475