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

Theorem sloteq 12001
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 5428 . . 3  |-  ( A  =  B  ->  (
f `  A )  =  ( f `  B ) )
21mpteq2dv 4026 . 2  |-  ( A  =  B  ->  (
f  e.  _V  |->  ( f `  A ) )  =  ( f  e.  _V  |->  ( f `
 B ) ) )
3 df-slot 12000 . 2  |- Slot  A  =  ( f  e.  _V  |->  ( f `  A
) )
4 df-slot 12000 . 2  |- Slot  B  =  ( f  e.  _V  |->  ( f `  B
) )
52, 3, 43eqtr4g 2198 1  |-  ( A  =  B  -> Slot  A  = Slot 
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   _Vcvv 2689    |-> cmpt 3996   ` cfv 5130  Slot cslot 11995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-iota 5095  df-fv 5138  df-slot 12000
This theorem is referenced by:  ndxid  12020
  Copyright terms: Public domain W3C validator