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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 B𝐶) = ∅

Proof of Theorem mpt20
Dummy variables w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mpt2 5460 . 2 (x ∅, y B𝐶) = {⟨⟨x, y⟩, z⟩ ∣ ((x y B) z = 𝐶)}
2 df-oprab 5459 . 2 {⟨⟨x, y⟩, z⟩ ∣ ((x y B) z = 𝐶)} = {wxyz(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 ∅)
53, 4mto 587 . . . . . 6 ¬ (w = ⟨⟨x, y⟩, z ((x y B) z = 𝐶))
65nex 1386 . . . . 5 ¬ z(w = ⟨⟨x, y⟩, z ((x y B) z = 𝐶))
76nex 1386 . . . 4 ¬ yz(w = ⟨⟨x, y⟩, z ((x y B) z = 𝐶))
87nex 1386 . . 3 ¬ xyz(w = ⟨⟨x, y⟩, z ((x y B) z = 𝐶))
98abf 3254 . 2 {wxyz(w = ⟨⟨x, y⟩, z ((x y B) z = 𝐶))} = ∅
101, 2, 93eqtri 2061 1 (x ∅, y B𝐶) = ∅
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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