Theorem of0r 32695

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 7697	. . . 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 5917	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4		dmeq 5917	. . . . . . 7 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
5	3, 4	ineqan12d 4230	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom ∅))
6	5	mpteq1d 5243	. . . . 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 481	. . . 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 5934	. . . . . . . 8 ⊢ dom ∅ = ∅
9	8	ineq2i 4225	. . . . . . 7 ⊢ (dom 𝐹 ∩ dom ∅) = (dom 𝐹 ∩ ∅)
10		in0 4401	. . . . . . 7 ⊢ (dom 𝐹 ∩ ∅) = ∅
11	9, 10	eqtri 2763	. . . . . 6 ⊢ (dom 𝐹 ∩ dom ∅) = ∅
12	11	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (dom 𝐹 ∩ dom ∅) = ∅)
13	12	mpteq1d 5243	. . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
14		mpt0 6711	. . . . 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 2779	. . 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 5313	. . . 4 ⊢ ∅ ∈ V
19	18	a1i 11	. . 3 ⊢ (𝐹 ∈ V → ∅ ∈ V)
20	2, 16, 17, 19, 19	ovmpod 7585	. 2 ⊢ (𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
21	1	reldmmpo 7567	. . 3 ⊢ Rel dom ∘f 𝑅
22	21	ovprc1 7470	. 2 ⊢ (¬ 𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
23	20, 22	pm2.61i 182	1 ⊢ (𝐹 ∘f 𝑅∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ∩ cin 3962  ∅c0 4339   ↦ cmpt 5231  dom cdm 5689  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   ∘_f cof 7695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697

This theorem is referenced by:  1arithidom  33545