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Theorem elbasov 13289
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 3536 . 2  |-  ( A  e.  B  ->  -.  B  =  (/) )
2 elbasov.s . . . . 5  |-  S  =  ( X O Y )
3 elbasov.o . . . . . 6  |-  Rel  dom  O
43ovprc 5972 . . . . 5  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  (/) )
52, 4syl5eq 2402 . . . 4  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  S  =  (/) )
65fveq2d 5612 . . 3  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
7 elbasov.b . . 3  |-  B  =  ( Base `  S
)
8 base0 13282 . . 3  |-  (/)  =  (
Base `  (/) )
96, 7, 83eqtr4g 2415 . 2  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  B  =  (/) )
101, 9nsyl2 119 1  |-  ( A  e.  B  ->  ( X  e.  _V  /\  Y  e.  _V ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710   _Vcvv 2864   (/)c0 3531   dom cdm 4771   Rel wrel 4776   ` cfv 5337  (class class class)co 5945   Basecbs 13245
This theorem is referenced by:  psrelbas  16224  psraddcl  16227  psrmulcllem  16231  psrvscafval  16234  psrvscacl  16237  resspsradd  16259  resspsrmul  16260  cphsubrglem  18717  mdegcl  19559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-slot 13249  df-base 13250
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