ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sloteq Unicode version

Theorem sloteq 11746
Description: Equality theorem for the Slot construction. The converse holds if  A (or  B) is a set. (Contributed by BJ, 27-Dec-2021.)
Assertion
Ref Expression
sloteq  |-  ( A  =  B  -> Slot  A  = Slot 
B )

Proof of Theorem sloteq
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq2 5353 . . 3  |-  ( A  =  B  ->  (
f `  A )  =  ( f `  B ) )
21mpteq2dv 3959 . 2  |-  ( A  =  B  ->  (
f  e.  _V  |->  ( f `  A ) )  =  ( f  e.  _V  |->  ( f `
 B ) ) )
3 df-slot 11745 . 2  |- Slot  A  =  ( f  e.  _V  |->  ( f `  A
) )
4 df-slot 11745 . 2  |- Slot  B  =  ( f  e.  _V  |->  ( f `  B
) )
52, 3, 43eqtr4g 2157 1  |-  ( A  =  B  -> Slot  A  = Slot 
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1299   _Vcvv 2641    |-> cmpt 3929   ` cfv 5059  Slot cslot 11740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-iota 5024  df-fv 5067  df-slot 11745
This theorem is referenced by:  ndxid  11765
  Copyright terms: Public domain W3C validator