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Theorem elbasov 14538
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 3790 . 2  |-  ( A  e.  B  ->  -.  B  =  (/) )
2 elbasov.s . . . . 5  |-  S  =  ( X O Y )
3 elbasov.o . . . . . 6  |-  Rel  dom  O
43ovprc 6311 . . . . 5  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( X O Y )  =  (/) )
52, 4syl5eq 2520 . . . 4  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  S  =  (/) )
65fveq2d 5870 . . 3  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  ( Base `  S
)  =  ( Base `  (/) ) )
7 elbasov.b . . 3  |-  B  =  ( Base `  S
)
8 base0 14529 . . 3  |-  (/)  =  (
Base `  (/) )
96, 7, 83eqtr4g 2533 . 2  |-  ( -.  ( X  e.  _V  /\  Y  e.  _V )  ->  B  =  (/) )
101, 9nsyl2 127 1  |-  ( A  e.  B  ->  ( X  e.  _V  /\  Y  e.  _V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   dom cdm 4999   Rel wrel 5004   ` cfv 5588  (class class class)co 6284   Basecbs 14490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-slot 14494  df-base 14495
This theorem is referenced by:  strov2rcl  14539  psrelbas  17831  psraddcl  17835  psrmulcllem  17839  psrvscafval  17842  psrvscacl  17845  resspsradd  17870  resspsrmul  17871  cphsubrglem  21387  mdegcl  22232
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