MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofeqd Structured version   Visualization version   GIF version

Theorem ofeqd 7699
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 7448 . . . 4 (𝜑 → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝑓𝑥)𝑆(𝑔𝑥)))
32mpteq2dv 5250 . . 3 (𝜑 → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
43mpoeq3dv 7512 . 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 7697 . 2 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
6 df-of 7697 . 2 f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
74, 5, 63eqtr4g 2800 1 (𝜑 → ∘f 𝑅 = ∘f 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Vcvv 3478  cin 3962  cmpt 5231  dom cdm 5689  cfv 6563  (class class class)co 7431  cmpo 7433  f cof 7695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697
This theorem is referenced by:  ofeq  7700  psrval  21953  resspsradd  22013  elrgspnlem1  33232  fedgmullem1  33657  fedgmullem2  33658  sitmval  34331  ldualset  39107  mendval  43168
  Copyright terms: Public domain W3C validator