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Theorem co01 5609
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 5494 . . . 4 ∅ = ∅
2 cnvco 5268 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 5242 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 5608 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2647 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2646 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 5257 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 5204 . . 3 Rel ∅
9 dfrel2 5542 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 220 . 2 ∅ = ∅
11 relco 5592 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 5542 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 220 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2652 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  c0 3891  ccnv 5073  ccom 5078  Rel wrel 5079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083
This theorem is referenced by:  xpcoid  5635  0trrel  13654  gsumval3  18229  utop2nei  21964  cononrel2  37379
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