Theorem sloteq 13167

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 5648	. . 3 |- ( A = B -> ( f ` A ) = ( f ` B ) )
2	1	mpteq2dv 4185	. 2 |- ( A = B -> ( f e. _V |-> ( f ` A ) ) = ( f e. _V |-> ( f ` B ) ) )
3		df-slot 13166	. 2 |- Slot A = ( f e. _V |-> ( f ` A ) )
4		df-slot 13166	. 2 |- Slot B = ( f e. _V |-> ( f ` B ) )
5	2, 3, 4	3eqtr4g 2289	1 |- ( A = B -> Slot A = Slot B )

Syntax hints:   -> wi 4   = wceq 1398   _Vcvv 2803   |-> cmpt 4155   ` cfv 5333  Slot cslot 13161

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-iota 5293  df-fv 5341  df-slot 13166

This theorem is referenced by:  ndxid  13186