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

Theorem ofeq 7064
Description: Equality theorem for function operation. (Contributed by Mario Carneiro, 20-Jul-2014.)
Assertion
Ref Expression
ofeq (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)

Proof of Theorem ofeq
Dummy variables 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1131 . . . . 5 ((𝑅 = 𝑆𝑓 ∈ V ∧ 𝑔 ∈ V) → 𝑅 = 𝑆)
21oveqd 6830 . . . 4 ((𝑅 = 𝑆𝑓 ∈ V ∧ 𝑔 ∈ V) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝑓𝑥)𝑆(𝑔𝑥)))
32mpteq2dv 4897 . . 3 ((𝑅 = 𝑆𝑓 ∈ V ∧ 𝑔 ∈ V) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
43mpt2eq3dva 6884 . 2 (𝑅 = 𝑆 → (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 7062 . 2 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
6 df-of 7062 . 2 𝑓 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
74, 5, 63eqtr4g 2819 1 (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  Vcvv 3340  cin 3714  cmpt 4881  dom cdm 5266  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑓 cof 7060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-iota 6012  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062
This theorem is referenced by:  psrval  19564  resspsradd  19618  resspsrvsca  19620  sitmval  30720  ldualset  34915  mendval  38255  mendplusgfval  38257  mendvscafval  38262
  Copyright terms: Public domain W3C validator