Theorem of0r 32635

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 7617	. . . 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 5850	. . . . . . 7 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
4		dmeq 5850	. . . . . . 7 ⊢ (𝑔 = ∅ → dom 𝑔 = dom ∅)
5	3, 4	ineqan12d 4175	. . . . . 6 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = ∅) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom ∅))
6	5	mpteq1d 5185	. . . . 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 5867	. . . . . . . 8 ⊢ dom ∅ = ∅
9	8	ineq2i 4170	. . . . . . 7 ⊢ (dom 𝐹 ∩ dom ∅) = (dom 𝐹 ∩ ∅)
10		in0 4348	. . . . . . 7 ⊢ (dom 𝐹 ∩ ∅) = ∅
11	9, 10	eqtri 2752	. . . . . 6 ⊢ (dom 𝐹 ∩ dom ∅) = ∅
12	11	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (dom 𝐹 ∩ dom ∅) = ∅)
13	12	mpteq1d 5185	. . . 4 ⊢ ((𝐹 ∈ V ∧ (𝑓 = 𝐹 ∧ 𝑔 = ∅)) → (𝑥 ∈ (dom 𝐹 ∩ dom ∅) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ ∅ ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
14		mpt0 6628	. . . . 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 2768	. . 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 5249	. . . 4 ⊢ ∅ ∈ V
19	18	a1i 11	. . 3 ⊢ (𝐹 ∈ V → ∅ ∈ V)
20	2, 16, 17, 19, 19	ovmpod 7505	. 2 ⊢ (𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
21	1	reldmmpo 7487	. . 3 ⊢ Rel dom ∘f 𝑅
22	21	ovprc1 7392	. 2 ⊢ (¬ 𝐹 ∈ V → (𝐹 ∘f 𝑅∅) = ∅)
23	20, 22	pm2.61i 182	1 ⊢ (𝐹 ∘f 𝑅∅) = ∅

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∩ cin 3904  ∅c0 4286   ↦ cmpt 5176  dom cdm 5623  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355   ∘_f cof 7615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617

This theorem is referenced by:  1arithidom  33484