ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  co01 Unicode version

Theorem co01 5123
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 5012 . . . 4  |-  `' (/)  =  (/)
2 cnvco 4794 . . . . 5  |-  `' (
(/)  o.  A )  =  ( `' A  o.  `' (/) )
31coeq2i 4769 . . . . 5  |-  ( `' A  o.  `' (/) )  =  ( `' A  o.  (/) )
4 co02 5122 . . . . 5  |-  ( `' A  o.  (/) )  =  (/)
52, 3, 43eqtri 2195 . . . 4  |-  `' (
(/)  o.  A )  =  (/)
61, 5eqtr4i 2194 . . 3  |-  `' (/)  =  `' ( (/)  o.  A
)
76cnveqi 4784 . 2  |-  `' `' (/)  =  `' `' (
(/)  o.  A )
8 rel0 4734 . . 3  |-  Rel  (/)
9 dfrel2 5059 . . 3  |-  ( Rel  (/) 
<->  `' `' (/)  =  (/) )
108, 9mpbi 144 . 2  |-  `' `' (/)  =  (/)
11 relco 5107 . . 3  |-  Rel  ( (/) 
o.  A )
12 dfrel2 5059 . . 3  |-  ( Rel  ( (/)  o.  A
)  <->  `' `' ( (/)  o.  A
)  =  ( (/)  o.  A ) )
1311, 12mpbi 144 . 2  |-  `' `' ( (/)  o.  A )  =  ( (/)  o.  A
)
147, 10, 133eqtr3ri 2200 1  |-  ( (/)  o.  A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1348   (/)c0 3414   `'ccnv 4608    o. ccom 4613   Rel wrel 4614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator