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Theorem ofceq 31596
 Description: Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Assertion
Ref Expression
ofceq (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)

Proof of Theorem ofceq
Dummy variables 𝑓 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq 7162 . . . 4 (𝑅 = 𝑆 → ((𝑓𝑥)𝑅𝑐) = ((𝑓𝑥)𝑆𝑐))
21mpteq2dv 5132 . . 3 (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
32mpoeq3dv 7233 . 2 (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))) = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐))))
4 df-ofc 31595 . 2 f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
5 df-ofc 31595 . 2 f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
63, 4, 53eqtr4g 2818 1 (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Vcvv 3409   ↦ cmpt 5116  dom cdm 5528  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158   ∘f/c cofc 31594 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-in 3867  df-ss 3877  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-iota 6299  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-ofc 31595 This theorem is referenced by: (None)
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