Theorem sloteq 12336

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 5480	. . 3 |- ( A = B -> ( f ` A ) = ( f ` B ) )
2	1	mpteq2dv 4067	. 2 |- ( A = B -> ( f e. _V |-> ( f ` A ) ) = ( f e. _V |-> ( f ` B ) ) )
3		df-slot 12335	. 2 |- Slot A = ( f e. _V |-> ( f ` A ) )
4		df-slot 12335	. 2 |- Slot B = ( f e. _V |-> ( f ` B ) )
5	2, 3, 4	3eqtr4g 2222	1 |- ( A = B -> Slot A = Slot B )

Syntax hints:   -> wi 4   = wceq 1342   _Vcvv 2721   |-> cmpt 4037   ` cfv 5182  Slot cslot 12330

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-iota 5147  df-fv 5190  df-slot 12335

This theorem is referenced by:  ndxid  12355