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Theorem co01 4863
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 4755 . . . 4 ∅ = ∅
2 cnvco 4548 . . . . 5 (∅ ∘ 𝐴) = (𝐴∅)
31coeq2i 4524 . . . . 5 (𝐴∅) = (𝐴 ∘ ∅)
4 co02 4862 . . . . 5 (𝐴 ∘ ∅) = ∅
52, 3, 43eqtri 2080 . . . 4 (∅ ∘ 𝐴) = ∅
61, 5eqtr4i 2079 . . 3 ∅ = (∅ ∘ 𝐴)
76cnveqi 4538 . 2 ∅ = (∅ ∘ 𝐴)
8 rel0 4490 . . 3 Rel ∅
9 dfrel2 4799 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 137 . 2 ∅ = ∅
11 relco 4847 . . 3 Rel (∅ ∘ 𝐴)
12 dfrel2 4799 . . 3 (Rel (∅ ∘ 𝐴) ↔ (∅ ∘ 𝐴) = (∅ ∘ 𝐴))
1311, 12mpbi 137 . 2 (∅ ∘ 𝐴) = (∅ ∘ 𝐴)
147, 10, 133eqtr3ri 2085 1 (∅ ∘ 𝐴) = ∅
Colors of variables: wff set class
Syntax hints:   = wceq 1259  c0 3252  ccnv 4372  ccom 4377  Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382
This theorem is referenced by: (None)
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