Theorem ofeqd 7622

Description: Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)

Hypothesis

Ref	Expression
ofeqd.1	⊢ (𝜑 → 𝑅 = 𝑆)

Assertion

Ref	Expression
ofeqd	⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆)

Proof of Theorem ofeqd

Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofeqd.1	. . . . 5 ⊢ (𝜑 → 𝑅 = 𝑆)
2	1	oveqd 7373	. . . 4 ⊢ (𝜑 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))
3	2	mpteq2dv 5190	. . 3 ⊢ (𝜑 → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
4	3	mpoeq3dv 7435	. 2 ⊢ (𝜑 → (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥)))) = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))))
5		df-of 7620	. 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6		df-of 7620	. 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
7	4, 5, 6	3eqtr4g 2794	1 ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆)

Syntax hints:   → wi 4   = wceq 1541  Vcvv 3438   ∩ cin 3898   ↦ cmpt 5177  dom cdm 5622  ‘cfv 6490  (class class class)co 7356   ∈ cmpo 7358   ∘_f cof 7618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-ss 3916  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-iota 6446  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620

This theorem is referenced by:  ofeq  7623  psrval  21869  resspsradd  21928  elrgspnlem1  33273  fedgmullem1  33735  fedgmullem2  33736  extdgfialglem1  33798  sitmval  34455  ldualset  39324  mendval  43363