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

Theorem co01 5567
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 5454 . . . 4 ∅ = ∅
2 cnvco 5230 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 5204 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 5566 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2636 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2635 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 5219 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 5166 . . 3 Rel ∅
9 dfrel2 5502 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 219 . 2 ∅ = ∅
11 relco 5550 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 5502 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 219 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2641 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  c0 3874  ccnv 5037  ccom 5042  Rel wrel 5043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047
This theorem is referenced by:  xpcoid  5593  0trrel  13568  gsumval3  18131  utop2nei  21864  cononrel2  36920
  Copyright terms: Public domain W3C validator