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Theorem co01 5386
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 5277 . . . 4  |-  `' (/)  =  (/)
2 cnvco 5058 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 5035 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 5385 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2462 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2461 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 5049 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 5001 . . 3  |-  Rel  (/)
9 dfrel2 5323 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 201 . 2  |-  `' `' (/)  =  (/)
11 relco 5370 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 5323 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 201 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2467 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1653   (/)c0 3630   `'ccnv 4879    o. ccom 4884   Rel wrel 4885
This theorem is referenced by:  xpcoid  5417  gsumval3  15516  utop2nei  18282
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889
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