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Theorem ofceq 31255
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 7151 . . . 4 (𝑅 = 𝑆 → ((𝑓𝑥)𝑅𝑐) = ((𝑓𝑥)𝑆𝑐))
21mpteq2dv 5153 . . 3 (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
32mpoeq3dv 7222 . 2 (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))) = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐))))
4 df-ofc 31254 . 2 f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
5 df-ofc 31254 . 2 f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
63, 4, 53eqtr4g 2878 1 (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  Vcvv 3492  cmpt 5137  dom cdm 5548  cfv 6348  (class class class)co 7145  cmpo 7147  f/c cofc 31253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ral 3140  df-rex 3141  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-iota 6307  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-ofc 31254
This theorem is referenced by: (None)
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