Theorem ofceq 34078

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 7437	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐))
2	1	mpteq2dv 5250	. . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
3	2	mpoeq3dv 7512	. 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 34077	. 2 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
5		df-ofc 34077	. 2 ⊢ ∘f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
6	3, 4, 5	3eqtr4g 2800	1 ⊢ (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)

Syntax hints:   → wi 4   = wceq 1537  Vcvv 3478   ↦ cmpt 5231  dom cdm 5689  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   ∘_f/c cofc 34076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ofc 34077

This theorem is referenced by: (None)