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

Theorem co01 5958
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 5844 . . . 4 ∅ = ∅
2 cnvco 5610 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 5585 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 5957 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2808 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2807 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 5599 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 5526 . . 3 Rel ∅
9 dfrel2 5891 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 222 . 2 ∅ = ∅
11 relco 5941 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 5891 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 222 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2813 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1508  c0 4181  ccnv 5410  ccom 5415  Rel wrel 5416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pr 5190
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-rab 3099  df-v 3419  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4182  df-if 4354  df-sn 4445  df-pr 4447  df-op 4451  df-br 4935  df-opab 4997  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420
This theorem is referenced by:  xpcoid  5984  0trrel  14208  gsumval3  18793  utop2nei  22577  cononrel2  39358
  Copyright terms: Public domain W3C validator