Theorem ofeqd 7727

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 7468	. . . 4 ⊢ (𝜑 → ((𝑓‘𝑥)𝑅(𝑔‘𝑥)) = ((𝑓‘𝑥)𝑆(𝑔‘𝑥)))
3	2	mpteq2dv 5205	. . 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 7530	. 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 7725	. 2 ⊢ ∘f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑅(𝑔‘𝑥))))
6		df-of 7725	. 2 ⊢ ∘f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓‘𝑥)𝑆(𝑔‘𝑥))))
7	4, 5, 6	3eqtr4g 2823	1 ⊢ (𝜑 → ∘f 𝑅 = ∘f 𝑆)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3454   ∩ cin 4092   ↦ cmpt 5192  dom cdm 5658  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   ∘_f cof 7723

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-iota 6520  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725

This theorem is used by:  ofeq  7728  psrval  22278  resspsradd  22337  elrgspnlem1  33902  fedgmullem1  34372  fedgmullem2  34373  extdgfialglem1  34435  sitmval  35092  ldualset  40448  mendval  44442