Theorem lmdfval 50118

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 7385	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 Func 𝑐) = (𝐷 Func 𝐶))
4	2	fveq2d 6845	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (oppCat‘𝑐) = (oppCat‘𝐶))
5	1, 2	oveq12d 7385	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑑 FuncCat 𝑐) = (𝐷 FuncCat 𝐶))
6	5	fveq2d 6845	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (oppCat‘(𝑑 FuncCat 𝑐)) = (oppCat‘(𝐷 FuncCat 𝐶)))
7	4, 6	oveq12d 7385	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐))) = ((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶))))
8		oveq12 7376	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝑐Δfunc𝑑) = (𝐶Δfunc𝐷))
9	8	fveq2d 6845	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → ( oppFunc ‘(𝑐Δfunc𝑑)) = ( oppFunc ‘(𝐶Δfunc𝐷)))
10		eqidd 2738	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → 𝑓 = 𝑓)
11	7, 9, 10	oveq123d 7388	. . . 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 5173	. . 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 50114	. . 3 ⊢ Limit = (𝑐 ∈ V, 𝑑 ∈ V ↦ (𝑓 ∈ (𝑑 Func 𝑐) ↦ (( oppFunc ‘(𝑐Δfunc𝑑))((oppCat‘𝑐) UP (oppCat‘(𝑑 FuncCat 𝑐)))𝑓)))
14		ovex 7400	. . . 4 ⊢ (𝐷 Func 𝐶) ∈ V
15	14	mptex 7178	. . 3 ⊢ (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) ∈ V
16	12, 13, 15	ovmpoa 7522	. 2 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Limit 𝐷) = (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)))
17		reldmlmd 50116	. . . 4 ⊢ Rel dom Limit
18	17	ovprc 7405	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 Limit 𝐷) = ∅)
19		ancom 460	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (𝐷 ∈ V ∧ 𝐶 ∈ V))
20		reldmfunc 49544	. . . . . . 7 ⊢ Rel dom Func
21	20	ovprc 7405	. . . . . 6 ⊢ (¬ (𝐷 ∈ V ∧ 𝐶 ∈ V) → (𝐷 Func 𝐶) = ∅)
22	19, 21	sylnbi 330	. . . . 5 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐷 Func 𝐶) = ∅)
23	22	mpteq1d 5176	. . . 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 6641	. . . 4 ⊢ (𝑓 ∈ ∅ ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) = ∅
25	23, 24	eqtrdi 2788	. . 3 ⊢ (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑓 ∈ (𝐷 Func 𝐶) ↦ (( oppFunc ‘(𝐶Δfunc𝐷))((oppCat‘𝐶) UP (oppCat‘(𝐷 FuncCat 𝐶)))𝑓)) = ∅)
26	18, 25	eqtr4d 2775	. 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 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274   ↦ cmpt 5167  ‘cfv 6499  (class class class)co 7367  oppCatcoppc 17677   Func cfunc 17821   FuncCat cfuc 17912  Δ_funccdiag 18178   oppFunc coppf 49591   UP cup 49642   Limit clmd 50112

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-lmd 50114

This theorem is referenced by:  lmdrcl  50120  reldmlmd2  50122  lmdfval2  50124  lmdpropd  50126