Theorem of0r 32546

Description: Function operation with the empty function. (Contributed by Thierry Arnoux, 27-May-2025.)

Assertion

Ref	Expression
of0r	⊢ (𝐹 ∘f 𝑅∅) = ∅

Proof of Theorem of0r

Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-of 7685	. . . 4 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
2	1	a1i 11	. . 3 ⊢ (𝐹 ∈ V → ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))))
3		dmeq 5906	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4		dmeq 5906	. . . . . . 7 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
5	3, 4	ineqan12d 4212	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom ∅))
6	5	mpteq1d 5244	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
7	6	adantl 480	. . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
8		dm0 5923	. . . . . . . 8 ⊢ dom ∅ = ∅
9	8	ineq2i 4207	. . . . . . 7 ⊢ (dom 𝐹 ∩ dom ∅) = (dom 𝐹 ∩ ∅)
10		in0 4393	. . . . . . 7 ⊢ (dom 𝐹 ∩ ∅) = ∅
11	9, 10	eqtri 2753	. . . . . 6 ⊢ (dom 𝐹 ∩ dom ∅) = ∅
12	11	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (dom 𝐹 ∩ dom ∅) = ∅)
13	12	mpteq1d 5244	. . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
14		mpt0 6698	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = ∅
15	14	a1i 11	. . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = ∅)
16	7, 13, 15	3eqtrd 2769	. . 3 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = ∅)
17		id 22	. . 3 ⊢ (𝐹 ∈ V → 𝐹 ∈ V)
18		0ex 5308	. . . 4 ⊢ ∅ ∈ V
19	18	a1i 11	. . 3 ⊢ (𝐹 ∈ V → ∅ ∈ V)
20	2, 16, 17, 19, 19	ovmpod 7573	. 2 ⊢ (𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
21	1	reldmmpo 7555	. . 3 ⊢ Rel dom ∘f 𝑅
22	21	ovprc1 7458	. 2 ⊢ (¬ 𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
23	20, 22	pm2.61i 182	1 ⊢ (𝐹 ∘f 𝑅∅) = ∅

Syntax hints:   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3461   ∩ cin 3943  ∅c0 4322   ↦ cmpt 5232  dom cdm 5678  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421   ∘_f cof 7683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fn 6552  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685

This theorem is referenced by:  1arithidom  33349