MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  co01 Structured version   Visualization version   GIF version

Theorem co01 6199
Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
co01 (∅ ∘ 𝐴) = ∅

Proof of Theorem co01
StepHypRef Expression
1 cnv0 6079 . . . 4 ∅ = ∅
2 cnvco 5827 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 5802 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 6198 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2768 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2767 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 5816 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 5741 . . 3 Rel ∅
9 dfrel2 6127 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 229 . 2 ∅ = ∅
11 relco 6046 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 6127 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 229 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2773 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  c0 4269  ccnv 5619  ccom 5624  Rel wrel 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-br 5093  df-opab 5155  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629
This theorem is referenced by:  xpcoid  6228  0trrel  14791  relexpsucrd  14843  relexpaddd  14864  gsumval3  19603  utop2nei  23508  cononrel2  41532
  Copyright terms: Public domain W3C validator