Theorem of0r 33061

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 7687	. . . 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 5898	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4		dmeq 5898	. . . . . . 7 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
5	3, 4	ineqan12d 4178	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom ∅))
6	5	mpteq1d 5206	. . . . 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 5915	. . . . . . . 8 ⊢ dom ∅ = ∅
9	8	ineq2i 4173	. . . . . . 7 ⊢ (dom 𝐹 ∩ dom ∅) = (dom 𝐹 ∩ ∅)
10		in0 4355	. . . . . . 7 ⊢ (dom 𝐹 ∩ ∅) = ∅
11	9, 10	eqtri 2789	. . . . . 6 ⊢ (dom 𝐹 ∩ dom ∅) = ∅
12	11	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (dom 𝐹 ∩ dom ∅) = ∅)
13	12	mpteq1d 5206	. . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
14		mpt0 6684	. . . . 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 2805	. . 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 5275	. . . 4 ⊢ ∅ ∈ V
19	18	a1i 11	. . 3 ⊢ (𝐹 ∈ V → ∅ ∈ V)
20	2, 16, 17, 19, 19	ovmpod 7575	. 2 ⊢ (𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
21	1	reldmmpo 7557	. . 3 ⊢ Rel dom ∘f 𝑅
22	21	ovprc1 7462	. 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 2146  Vcvv 3458   ∩ cin 3907  ∅c0 4289   ↦ cmpt 5197  dom cdm 5666  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425   ∘_f cof 7685

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fn 6546  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-of 7687

This theorem is used by:  1arithidom  33858