Theorem sloteq 12623

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 5554	. . 3 |- ( A = B -> ( f ` A ) = ( f ` B ) )
2	1	mpteq2dv 4120	. 2 |- ( A = B -> ( f e. _V |-> ( f ` A ) ) = ( f e. _V |-> ( f ` B ) ) )
3		df-slot 12622	. 2 |- Slot A = ( f e. _V |-> ( f ` A ) )
4		df-slot 12622	. 2 |- Slot B = ( f e. _V |-> ( f ` B ) )
5	2, 3, 4	3eqtr4g 2251	1 |- ( A = B -> Slot A = Slot B )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2760   |-> cmpt 4090   ` cfv 5254  Slot cslot 12617

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 2175

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-iota 5215  df-fv 5262  df-slot 12622

This theorem is referenced by:  ndxid  12642