Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sloteq Structured version   Visualization version   GIF version

Theorem sloteq 16546
 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 6658 . . 3 (𝐴 = 𝐵 → (𝑓𝐴) = (𝑓𝐵))
21mpteq2dv 5128 . 2 (𝐴 = 𝐵 → (𝑓 ∈ V ↦ (𝑓𝐴)) = (𝑓 ∈ V ↦ (𝑓𝐵)))
3 df-slot 16545 . 2 Slot 𝐴 = (𝑓 ∈ V ↦ (𝑓𝐴))
4 df-slot 16545 . 2 Slot 𝐵 = (𝑓 ∈ V ↦ (𝑓𝐵))
52, 3, 43eqtr4g 2818 1 (𝐴 = 𝐵 → Slot 𝐴 = Slot 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Vcvv 3409   ↦ cmpt 5112  ‘cfv 6335  Slot cslot 16540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-iota 6294  df-fv 6343  df-slot 16545 This theorem is referenced by:  ndxid  16567
 Copyright terms: Public domain W3C validator