MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ovima0 Structured version   Unicode version

Theorem ovima0 6453
Description: An operation value is a member of the image plus null. (Contributed by Thierry Arnoux, 25-Jun-2019.)
Assertion
Ref Expression
ovima0  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X R Y )  e.  ( ( R " ( A  X.  B ) )  u.  { (/) } ) )

Proof of Theorem ovima0
StepHypRef Expression
1 simpr 461 . . 3  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  ( X R Y )  =  (/) )  ->  ( X R Y )  =  (/) )
2 ssun2 3664 . . . 4  |-  { (/) } 
C_  ( ( R
" ( A  X.  B ) )  u. 
{ (/) } )
3 0ex 4587 . . . . 5  |-  (/)  e.  _V
43snid 4060 . . . 4  |-  (/)  e.  { (/)
}
52, 4sselii 3496 . . 3  |-  (/)  e.  ( ( R " ( A  X.  B ) )  u.  { (/) } )
61, 5syl6eqel 2553 . 2  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  ( X R Y )  =  (/) )  ->  ( X R Y )  e.  ( ( R " ( A  X.  B ) )  u.  { (/) } ) )
7 ssun1 3663 . . 3  |-  ( R
" ( A  X.  B ) )  C_  ( ( R "
( A  X.  B
) )  u.  { (/)
} )
8 df-ov 6299 . . . 4  |-  ( X R Y )  =  ( R `  <. X ,  Y >. )
9 opelxpi 5040 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  B )  -> 
<. X ,  Y >.  e.  ( A  X.  B
) )
108eqeq1i 2464 . . . . . . 7  |-  ( ( X R Y )  =  (/)  <->  ( R `  <. X ,  Y >. )  =  (/) )
1110notbii 296 . . . . . 6  |-  ( -.  ( X R Y )  =  (/)  <->  -.  ( R `  <. X ,  Y >. )  =  (/) )
1211biimpi 194 . . . . 5  |-  ( -.  ( X R Y )  =  (/)  ->  -.  ( R `  <. X ,  Y >. )  =  (/) )
13 eliman0 5901 . . . . 5  |-  ( (
<. X ,  Y >.  e.  ( A  X.  B
)  /\  -.  ( R `  <. X ,  Y >. )  =  (/) )  ->  ( R `  <. X ,  Y >. )  e.  ( R "
( A  X.  B
) ) )
149, 12, 13syl2an 477 . . . 4  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  -.  ( X R Y )  =  (/) )  ->  ( R `
 <. X ,  Y >. )  e.  ( R
" ( A  X.  B ) ) )
158, 14syl5eqel 2549 . . 3  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  -.  ( X R Y )  =  (/) )  ->  ( X R Y )  e.  ( R " ( A  X.  B ) ) )
167, 15sseldi 3497 . 2  |-  ( ( ( X  e.  A  /\  Y  e.  B
)  /\  -.  ( X R Y )  =  (/) )  ->  ( X R Y )  e.  ( ( R "
( A  X.  B
) )  u.  { (/)
} ) )
176, 16pm2.61dan 791 1  |-  ( ( X  e.  A  /\  Y  e.  B )  ->  ( X R Y )  e.  ( ( R " ( A  X.  B ) )  u.  { (/) } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819    u. cun 3469   (/)c0 3793   {csn 4032   <.cop 4038    X. cxp 5006   "cima 5011   ` cfv 5594  (class class class)co 6296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fv 5602  df-ov 6299
This theorem is referenced by:  legval  24097
  Copyright terms: Public domain W3C validator