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Theorem ofceq 31032
Description: Equality theorem for function/constant operation. (Contributed by Thierry Arnoux, 30-Jan-2017.)
Assertion
Ref Expression
ofceq (𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆)

Proof of Theorem ofceq
Dummy variables 𝑓 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq 6980 . . . 4 (𝑅 = 𝑆 → ((𝑓𝑥)𝑅𝑐) = ((𝑓𝑥)𝑆𝑐))
21mpteq2dv 5019 . . 3 (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
32mpoeq3dv 7049 . 2 (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))) = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐))))
4 df-ofc 31031 . 2 𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
5 df-ofc 31031 . 2 𝑓/𝑐𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
63, 4, 53eqtr4g 2832 1 (𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1508  Vcvv 3408  cmpt 5004  dom cdm 5403  cfv 6185  (class class class)co 6974  cmpo 6976  𝑓/𝑐cofc 31030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-12 2107  ax-ext 2743
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-ral 3086  df-rex 3087  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-iota 6149  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-ofc 31031
This theorem is referenced by: (None)
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