Theorem sloteq 11964

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.

Step	Hyp	Ref	Expression
1		fveq2 5421	. . 3 |- ( A = B -> ( f ` A ) = ( f ` B ) )
2	1	mpteq2dv 4019	. 2 |- ( A = B -> ( f e. _V |-> ( f ` A ) ) = ( f e. _V |-> ( f ` B ) ) )
3		df-slot 11963	. 2 |- Slot A = ( f e. _V |-> ( f ` A ) )
4		df-slot 11963	. 2 |- Slot B = ( f e. _V |-> ( f ` B ) )
5	2, 3, 4	3eqtr4g 2197	1 |- ( A = B -> Slot A = Slot B )

Syntax hints:   -> wi 4   = wceq 1331   _Vcvv 2686   |-> cmpt 3989   ` cfv 5123  Slot cslot 11958

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-iota 5088  df-fv 5131  df-slot 11963

This theorem is referenced by:  ndxid  11983