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.

Step	Hyp	Ref	Expression
1		oveq 6555	. . . 4 ⊢ (𝑅 = 𝑆 → ((𝑓‘𝑥)𝑅𝑐) = ((𝑓‘𝑥)𝑆𝑐))
2	1	mpteq2dv 4673	. . 3 ⊢ (𝑅 = 𝑆 → (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)) = (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
3	2	mpt2eq3dv 6619	. 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 29485	. 2 ⊢ ∘𝑓/𝑐𝑅 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑅𝑐)))
5		df-ofc 29485	. 2 ⊢ ∘𝑓/𝑐𝑆 = (𝑓 ∈ V, 𝑐 ∈ V ↦ (𝑥 ∈ dom 𝑓 ↦ ((𝑓‘𝑥)𝑆𝑐)))
6	3, 4, 5	3eqtr4g 2669	1 ⊢ (𝑅 = 𝑆 → ∘𝑓/𝑐𝑅 = ∘𝑓/𝑐𝑆)

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)