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Theorem co01 5811
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 5693 . . . 4 ∅ = ∅
2 cnvco 5463 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 5438 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 5810 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2786 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2785 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 5452 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 5399 . . 3 Rel ∅
9 dfrel2 5741 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 220 . 2 ∅ = ∅
11 relco 5794 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 5741 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 220 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2791 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  c0 4058  ccnv 5265  ccom 5270  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275
This theorem is referenced by:  xpcoid  5837  0trrel  13921  gsumval3  18508  utop2nei  22255  cononrel2  38403
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