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Theorem co01 6281
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 6160 . . . 4 ∅ = ∅
2 cnvco 5896 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 5871 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 6280 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2769 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2768 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 5885 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 5809 . . 3 Rel ∅
9 dfrel2 6209 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 230 . 2 ∅ = ∅
11 relco 6126 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 6209 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 230 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2774 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  c0 4333  ccnv 5684  ccom 5689  Rel wrel 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694
This theorem is referenced by:  xpcoid  6310  0trrel  15020  relexpsucrd  15072  relexpaddd  15093  gsumval3  19925  utop2nei  24259  cononrel2  43608
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