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Theorem co01 5023
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 4912 . . . 4  |-  `' (/)  =  (/)
2 cnvco 4694 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 4669 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 5022 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2142 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2141 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 4684 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 4634 . . 3  |-  Rel  (/)
9 dfrel2 4959 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 144 . 2  |-  `' `' (/)  =  (/)
11 relco 5007 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 4959 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 144 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2147 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1316   (/)c0 3333   `'ccnv 4508    o. ccom 4513   Rel wrel 4514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518
This theorem is referenced by: (None)
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