Theorem ofeqd 7677

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 7428	. . . 4 ⊢ (𝜑 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))
3	2	mpteq2dv 5209	. . 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 7490	. 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 7675	. 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6		df-of 7675	. 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
7	4, 5, 6	3eqtr4g 2829	1 ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆)

Syntax hints:   → wi 4   = wceq 1567  Vcvv 3463   ∩ cin 3912   ↦ cmpt 5196  dom cdm 5662  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   ∘_f cof 7673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-ss 3930  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-iota 6493  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675

This theorem is referenced by:  ofeq  7678  psrval  22033  resspsradd  22092  elrgspnlem1  33502  fedgmullem1  33963  fedgmullem2  33964  extdgfialglem1  34026  sitmval  34683  ldualset  39788  mendval  43797