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Theorem co01 6092
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 5977 . . . 4 ∅ = ∅
2 cnvco 5733 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 5708 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 6091 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2849 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2848 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 5722 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 5649 . . 3 Rel ∅
9 dfrel2 6024 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 233 . 2 ∅ = ∅
11 relco 6075 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 6024 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 233 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2854 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  c0 4265  ccnv 5531  ccom 5536  Rel wrel 5537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541
This theorem is referenced by:  xpcoid  6119  0trrel  14332  gsumval3  19018  utop2nei  22854  cononrel2  40226
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