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Theorem elbasov 13192
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 3460 . 2  |-  ( A  e.  B  ->  -.  B  =  (/) )
2 elbasov.s . . . . 5  |-  S  =  ( X O Y )
3 elbasov.o . . . . . 6  |-  Rel  dom  O
43ovprc 5885 . . . . 5  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  (/) )
52, 4syl5eq 2327 . . . 4  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  S  =  (/) )
65fveq2d 5529 . . 3  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
7 elbasov.b . . 3  |-  B  =  ( Base `  S
)
8 base0 13185 . . 3  |-  (/)  =  (
Base `  (/) )
96, 7, 83eqtr4g 2340 . 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 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   dom cdm 4689   Rel wrel 4694   ` cfv 5255  (class class class)co 5858   Basecbs 13148
This theorem is referenced by:  psrelbas  16125  psraddcl  16128  psrmulcllem  16132  psrvscafval  16135  psrvscacl  16138  resspsradd  16160  resspsrmul  16161  cphsubrglem  18613  mdegcl  19455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-slot 13152  df-base 13153
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