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Theorem ofceq 33716
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 7426 . . . 4 (𝑅 = 𝑆 → ((𝑓𝑥)𝑅𝑐) = ((𝑓𝑥)𝑆𝑐))
21mpteq2dv 5250 . . 3 (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
32mpoeq3dv 7499 . 2 (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))) = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐))))
4 df-ofc 33715 . 2 f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
5 df-ofc 33715 . 2 f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
63, 4, 53eqtr4g 2793 1 (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  Vcvv 3471  cmpt 5231  dom cdm 5678  cfv 6548  (class class class)co 7420  cmpo 7422  f/c cofc 33714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-in 3954  df-ss 3964  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6500  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-ofc 33715
This theorem is referenced by: (None)
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