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Theorem ofeq 5896
 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 946 . . . . 5 ((𝑅 = 𝑆𝑓 ∈ V ∧ 𝑔 ∈ V) → 𝑅 = 𝑆)
21oveqd 5707 . . . 4 ((𝑅 = 𝑆𝑓 ∈ V ∧ 𝑔 ∈ V) → ((𝑓𝑥)𝑅(𝑔𝑥)) = ((𝑓𝑥)𝑆(𝑔𝑥)))
32mpteq2dv 3951 . . 3 ((𝑅 = 𝑆𝑓 ∈ V ∧ 𝑔 ∈ V) → (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))) = (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
43mpt2eq3dva 5751 . 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 5894 . 2 𝑓 𝑅 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑅(𝑔𝑥))))
6 df-of 5894 . 2 𝑓 𝑆 = (𝑓 ∈ V, 𝑔 ∈ V ↦ (𝑥 ∈ (dom 𝑓 ∩ dom 𝑔) ↦ ((𝑓𝑥)𝑆(𝑔𝑥))))
74, 5, 63eqtr4g 2152 1 (𝑅 = 𝑆 → ∘𝑓 𝑅 = ∘𝑓 𝑆)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ w3a 927   = wceq 1296   ∈ wcel 1445  Vcvv 2633   ∩ cin 3012   ↦ cmpt 3921  dom cdm 4467  ‘cfv 5049  (class class class)co 5690   ↦ cmpt2 5692   ∘𝑓 cof 5892 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-iota 5014  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-of 5894 This theorem is referenced by: (None)
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