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Theorem co01 5505
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 5394 . . . 4  |-  `' (/)  =  (/)
2 cnvco 5177 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 5152 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 5504 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2487 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2486 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 5166 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 5115 . . 3  |-  Rel  (/)
9 dfrel2 5441 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 208 . 2  |-  `' `' (/)  =  (/)
11 relco 5488 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 5441 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 208 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2492 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   (/)c0 3783   `'ccnv 4987    o. ccom 4992   Rel wrel 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997
This theorem is referenced by:  xpcoid  5531  0trrel  12899  gsumval3OLD  17107  gsumval3  17110  utop2nei  20919
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