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Theorem sloteq 12520
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 5534 . . 3  |-  ( A  =  B  ->  (
f `  A )  =  ( f `  B ) )
21mpteq2dv 4109 . 2  |-  ( A  =  B  ->  (
f  e.  _V  |->  ( f `  A ) )  =  ( f  e.  _V  |->  ( f `
 B ) ) )
3 df-slot 12519 . 2  |- Slot  A  =  ( f  e.  _V  |->  ( f `  A
) )
4 df-slot 12519 . 2  |- Slot  B  =  ( f  e.  _V  |->  ( f `  B
) )
52, 3, 43eqtr4g 2247 1  |-  ( A  =  B  -> Slot  A  = Slot 
B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364   _Vcvv 2752    |-> cmpt 4079   ` cfv 5235  Slot cslot 12514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-iota 5196  df-fv 5243  df-slot 12519
This theorem is referenced by:  ndxid  12539
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