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

Theorem ofeqd 7555
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 7312 . . . 4 (𝜑 → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝑓𝑥)𝑆(𝑔𝑥)))
32mpteq2dv 5179 . . 3 (𝜑 → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
43mpoeq3dv 7374 . 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 7553 . 2 f 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
6 df-of 7553 . 2 f 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
74, 5, 63eqtr4g 2798 1 (𝜑 → ∘f 𝑅 = ∘f 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  Vcvv 3434  cin 3888  cmpt 5160  dom cdm 5591  cfv 6447  (class class class)co 7295  cmpo 7297  f cof 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1540  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3436  df-in 3896  df-ss 3906  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-iota 6399  df-fv 6455  df-ov 7298  df-oprab 7299  df-mpo 7300  df-of 7553
This theorem is referenced by:  ofeq  7556  psrval  21146  resspsradd  21213  fedgmullem1  31738  fedgmullem2  31739  sitmval  32344  ldualset  37165  mendval  41032
  Copyright terms: Public domain W3C validator