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Mirrors > Home > ILE Home > Th. List > mpt20 | GIF version |
Description: A mapping operation with empty domain. (Contributed by Stefan O'Rear, 29-Jan-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
mpt20 | ⊢ (x ∈ ∅, y ∈ B ↦ 𝐶) = ∅ |
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) = ∅ |
Colors of variables: wff set class |
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) |
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