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Theorem co01 5203
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 5100 . . . 4  |-  `' (/)  =  (/)
2 cnvco 4881 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 4860 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 5202 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2320 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2319 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 4872 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 4826 . . 3  |-  Rel  (/)
9 dfrel2 5140 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 199 . 2  |-  `' `' (/)  =  (/)
11 relco 5187 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 5140 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 199 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2325 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1632   (/)c0 3468   `'ccnv 4704    o. ccom 4709   Rel wrel 4710
This theorem is referenced by:  gsumval3  15207  empos  25345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714
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