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Theorem co01 4865
Description: Composition with the empty set. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
co01  |-  ( (/)  o.  A )  =  (/)

Proof of Theorem co01
StepHypRef Expression
1 cnv0 4757 . . . 4  |-  `' (/)  =  (/)
2 cnvco 4548 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 4524 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 4864 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2106 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2105 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 4538 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 4490 . . 3  |-  Rel  (/)
9 dfrel2 4801 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 143 . 2  |-  `' `' (/)  =  (/)
11 relco 4849 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 4801 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 143 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2111 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1285   (/)c0 3258   `'ccnv 4370    o. ccom 4375   Rel wrel 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380
This theorem is referenced by: (None)
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