Theorem of0r 33255

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 7682	. . . 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 5885	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4		dmeq 5885	. . . . . . 7 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
5	3, 4	ineqan12d 4168	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom ∅))
6	5	mpteq1d 5195	. . . . 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 487	. . . 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 5902	. . . . . . . 8 ⊢ dom ∅ = ∅
9	8	ineq2i 4163	. . . . . . 7 ⊢ (dom 𝐹 ∩ dom ∅) = (dom 𝐹 ∩ ∅)
10		in0 4345	. . . . . . 7 ⊢ (dom 𝐹 ∩ ∅) = ∅
11	9, 10	eqtri 2784	. . . . . 6 ⊢ (dom 𝐹 ∩ dom ∅) = ∅
12	11	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (dom 𝐹 ∩ dom ∅) = ∅)
13	12	mpteq1d 5195	. . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
14		mpt0 6673	. . . . 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 2800	. . 3 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = ∅)
17		id 23	. . 3 ⊢ (𝐹 ∈ V → 𝐹 ∈ V)
18		0ex 5261	. . . 4 ⊢ ∅ ∈ V
19	18	a1i 11	. . 3 ⊢ (𝐹 ∈ V → ∅ ∈ V)
20	2, 16, 17, 19, 19	ovmpod 7564	. 2 ⊢ (𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
21	1	reldmmpo 7546	. . 3 ⊢ Rel dom ∘f 𝑅
22	21	ovprc1 7451	. 2 ⊢ (¬ 𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
23	20, 22	pm2.61i 184	1 ⊢ (𝐹 ∘f 𝑅∅) = ∅

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898  ∅c0 4279   ↦ cmpt 5186  dom cdm 5651  ‘cfv 6531  (class class class)co 7412   ∈ cmpo 7414   ∘_f cof 7680

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fn 6534  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7682

This theorem is used by:  1arithidom  34051