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Theorem co01 5522
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 5409 . . . 4  |-  `' (/)  =  (/)
2 cnvco 5188 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 5163 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 5521 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2500 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2499 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 5177 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 5127 . . 3  |-  Rel  (/)
9 dfrel2 5457 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 208 . 2  |-  `' `' (/)  =  (/)
11 relco 5505 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 5457 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 208 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2505 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   (/)c0 3785   `'ccnv 4998    o. ccom 5003   Rel wrel 5004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008
This theorem is referenced by:  xpcoid  5548  gsumval3OLD  16711  gsumval3  16714  utop2nei  20516  0trrel  36804
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