Theorem mpt20 5516

Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
mpt20	⊢ (x ∈ ∅, y ∈ B ↦ 𝐶) = ∅

Proof of Theorem mpt20

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt2 5460	. 2 ⊢ (x ∈ ∅, y ∈ B ↦ 𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶)}
2		df-oprab 5459	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶)} = {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶))}
3		noel 3222	. . . . . . 7 ⊢ ¬ x ∈ ∅
4		simprll 489	. . . . . . 7 ⊢ ((w = ⟨⟨x, y⟩, z⟩ ∧ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶)) → x ∈ ∅)
5	3, 4	mto 587	. . . . . 6 ⊢ ¬ (w = ⟨⟨x, y⟩, z⟩ ∧ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶))
6	5	nex 1386	. . . . 5 ⊢ ¬ ∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶))
7	6	nex 1386	. . . 4 ⊢ ¬ ∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶))
8	7	nex 1386	. . 3 ⊢ ¬ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶))
9	8	abf 3254	. 2 ⊢ {w ∣ ∃x∃y∃z(w = ⟨⟨x, y⟩, z⟩ ∧ ((x ∈ ∅ ∧ y ∈ B) ∧ z = 𝐶))} = ∅
10	1, 2, 9	3eqtri 2061	1 ⊢ (x ∈ ∅, y ∈ B ↦ 𝐶) = ∅

Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∅c0 3218  ⟨cop 3370  {coprab 5456   ↦ cmpt2 5457

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-nul 3219  df-oprab 5459  df-mpt2 5460

This theorem is referenced by: (None)