Theorem ofceq 33083

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 7411	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐))
2	1	mpteq2dv 5249	. . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
3	2	mpoeq3dv 7484	. 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 33082	. 2 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
5		df-ofc 33082	. 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 1541  Vcvv 3474   ↦ cmpt 5230  dom cdm 5675  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ∘_f/c cofc 33081

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-in 3954  df-ss 3964  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-iota 6492  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-ofc 33082

This theorem is referenced by: (None)