Theorem ofceq 34094

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 7396	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐))
2	1	mpteq2dv 5204	. . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
3	2	mpoeq3dv 7471	. 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 34093	. 2 ⊢ ∘f/c 𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
5		df-ofc 34093	. 2 ⊢ ∘f/c 𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
6	3, 4, 5	3eqtr4g 2790	1 ⊢ (𝑅 = 𝑆 → ∘f/c 𝑅 = ∘f/c 𝑆)

Syntax hints:   → wi 4   = wceq 1540  Vcvv 3450   ↦ cmpt 5191  dom cdm 5641  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ∘_f/c cofc 34092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-ss 3934  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-iota 6467  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-ofc 34093

This theorem is referenced by: (None)