MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elbasov Structured version   Unicode version

Theorem elbasov 13513
Description: Utility theorem: reverse closure for any structure defined as a two-argument function. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
elbasov.o  |-  Rel  dom  O
elbasov.s  |-  S  =  ( X O Y )
elbasov.b  |-  B  =  ( Base `  S
)
Assertion
Ref Expression
elbasov  |-  ( A  e.  B  ->  ( X  e.  _V  /\  Y  e.  _V ) )

Proof of Theorem elbasov
StepHypRef Expression
1 n0i 3633 . 2  |-  ( A  e.  B  ->  -.  B  =  (/) )
2 elbasov.s . . . . 5  |-  S  =  ( X O Y )
3 elbasov.o . . . . . 6  |-  Rel  dom  O
43ovprc 6108 . . . . 5  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  (/) )
52, 4syl5eq 2480 . . . 4  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  S  =  (/) )
65fveq2d 5732 . . 3  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
7 elbasov.b . . 3  |-  B  =  ( Base `  S
)
8 base0 13506 . . 3  |-  (/)  =  (
Base `  (/) )
96, 7, 83eqtr4g 2493 . 2  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  B  =  (/) )
101, 9nsyl2 121 1  |-  ( A  e.  B  ->  ( X  e.  _V  /\  Y  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956   (/)c0 3628   dom cdm 4878   Rel wrel 4883   ` cfv 5454  (class class class)co 6081   Basecbs 13469
This theorem is referenced by:  psrelbas  16444  psraddcl  16447  psrmulcllem  16451  psrvscafval  16454  psrvscacl  16457  resspsradd  16479  resspsrmul  16480  cphsubrglem  19140  mdegcl  19992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-slot 13473  df-base 13474
  Copyright terms: Public domain W3C validator