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Theorem 2trllemF 24756
Description: Lemma 5 for constr2trl 24806. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
Assertion
Ref Expression
2trllemF  |-  ( ( ( E `  I
)  =  { X ,  Y }  /\  Y  e.  V )  ->  I  e.  dom  E )

Proof of Theorem 2trllemF
StepHypRef Expression
1 prid2g 4123 . . . 4  |-  ( Y  e.  V  ->  Y  e.  { X ,  Y } )
2 eleq2 2527 . . . 4  |-  ( ( E `  I )  =  { X ,  Y }  ->  ( Y  e.  ( E `  I )  <->  Y  e.  { X ,  Y }
) )
31, 2syl5ibr 221 . . 3  |-  ( ( E `  I )  =  { X ,  Y }  ->  ( Y  e.  V  ->  Y  e.  ( E `  I
) ) )
43imp 427 . 2  |-  ( ( ( E `  I
)  =  { X ,  Y }  /\  Y  e.  V )  ->  Y  e.  ( E `  I
) )
5 elfvdm 5874 . 2  |-  ( Y  e.  ( E `  I )  ->  I  e.  dom  E )
64, 5syl 16 1  |-  ( ( ( E `  I
)  =  { X ,  Y }  /\  Y  e.  V )  ->  I  e.  dom  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   {cpr 4018   dom cdm 4988   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-nul 4568  ax-pow 4615
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-dm 4998  df-iota 5534  df-fv 5578
This theorem is referenced by:  2trllemH  24759  2trllemE  24760
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