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Theorem ofeqd 7530
Description: Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)
Hypothesis
Ref Expression
ofeqd.1 (𝜑𝑅 = 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.
StepHypRef Expression
1 ofeqd.1 . . . . 5 (𝜑𝑅 = 𝑆)
21oveqd 7289 . . . 4 (𝜑 → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝑓𝑥)𝑆(𝑔𝑥)))
32mpteq2dv 5181 . . 3 (𝜑 → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
43mpoeq3dv 7349 . 2 (𝜑 → (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥)))) = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥)))))
5 df-of 7528 . 2 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
6 df-of 7528 . 2 f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
74, 5, 63eqtr4g 2805 1 (𝜑 → ∘f 𝑅 = ∘f 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  Vcvv 3431  cin 3891  cmpt 5162  dom cdm 5590  cfv 6432  (class class class)co 7272  cmpo 7274  f cof 7526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711
This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1545  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-in 3899  df-ss 3909  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5163  df-iota 6390  df-fv 6440  df-ov 7275  df-oprab 7276  df-mpo 7277  df-of 7528
This theorem is referenced by:  ofeq  7531  psrval  21129  resspsradd  21196  fedgmullem1  31719  fedgmullem2  31720  sitmval  32325  ldualset  37148  mendval  41017
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