Theorem ofceq 34281

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.

Step	Hyp	Ref	Expression
1		oveq 7376	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐))
2	1	mpteq2dv 5194	. . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
3	2	mpoeq3dv 7449	. 2 ⊢ (𝑅 = 𝑆 → (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐))) = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐))))
4		df-ofc 34280	. 2 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
5		df-ofc 34280	. 2 ⊢ ∘f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
6	3, 4, 5	3eqtr4g 2797	1 ⊢ (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)

Syntax hints:   → wi 4   = wceq 1542  Vcvv 3442   ↦ cmpt 5181  dom cdm 5634  ‘cfv 6502  (class class class)co 7370   ∈ cmpo 7372   ∘_f/c cofc 34279

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-ss 3920  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-iota 6458  df-fv 6510  df-ov 7373  df-oprab 7374  df-mpo 7375  df-ofc 34280

This theorem is referenced by: (None)