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Theorem co01 4835
 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 4727 . . . 4 ∅ = ∅
2 cnvco 4520 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 4496 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 4834 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2064 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2063 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 4510 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 4462 . . 3 Rel ∅
9 dfrel2 4771 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 133 . 2 ∅ = ∅
11 relco 4819 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 4771 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 133 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2069 1 (∅ ∘ 𝐴) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1243  ∅c0 3224  ◡ccnv 4344   ∘ ccom 4349  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354 This theorem is referenced by: (None)
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