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Theorem co01 4778
 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 4670 . . . 4 ∅ = ∅
2 cnvco 4463 . . . . 5 (∅ ∘ A) = (A∅)
31coeq2i 4439 . . . . 5 (A∅) = (A ∘ ∅)
4 co02 4777 . . . . 5 (A ∘ ∅) = ∅
52, 3, 43eqtri 2061 . . . 4 (∅ ∘ A) = ∅
61, 5eqtr4i 2060 . . 3 ∅ = (∅ ∘ A)
76cnveqi 4453 . 2 ∅ = (∅ ∘ A)
8 rel0 4405 . . 3 Rel ∅
9 dfrel2 4714 . . 3 (Rel ∅ ↔ ∅ = ∅)
108, 9mpbi 133 . 2 ∅ = ∅
11 relco 4762 . . 3 Rel (∅ ∘ A)
12 dfrel2 4714 . . 3 (Rel (∅ ∘ A) ↔ (∅ ∘ A) = (∅ ∘ A))
1311, 12mpbi 133 . 2 (∅ ∘ A) = (∅ ∘ A)
147, 10, 133eqtr3ri 2066 1 (∅ ∘ A) = ∅
 Colors of variables: wff set class Syntax hints:   = wceq 1242  ∅c0 3218  ◡ccnv 4287   ∘ ccom 4292  Rel wrel 4293 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297 This theorem is referenced by: (None)
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