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Theorem ofceq 29486
 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 6555 . . . 4 (𝑅 = 𝑆 → ((𝑓𝑥)𝑅𝑐) = ((𝑓𝑥)𝑆𝑐))
21mpteq2dv 4673 . . 3 (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
32mpt2eq3dv 6619 . 2 (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐))) = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐))))
4 df-ofc 29485 . 2 𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑅𝑐)))
5 df-ofc 29485 . 2 𝑓/𝑐𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓𝑥)𝑆𝑐)))
63, 4, 53eqtr4g 2669 1 (𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475  Vcvv 3173   ↦ cmpt 4643  dom cdm 5038  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  ∘𝑓/𝑐cofc 29484 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ofc 29485 This theorem is referenced by: (None)
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