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Theorem co01 5187
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 5084 . . . 4  |-  `' (/)  =  (/)
2 cnvco 4865 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 4844 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 5186 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2307 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2306 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 4856 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 4810 . . 3  |-  Rel  (/)
9 dfrel2 5124 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 199 . 2  |-  `' `' (/)  =  (/)
11 relco 5171 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 5124 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 199 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2312 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1623   (/)c0 3455   `'ccnv 4688    o. ccom 4693   Rel wrel 4694
This theorem is referenced by:  gsumval3  15191  empos  25242
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698
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