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Theorem tendo0cbv 35893
 Description: Define additive identity for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)
Hypothesis
Ref Expression
tendo0cbv.o 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
Assertion
Ref Expression
tendo0cbv 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
Distinct variable groups:   𝐵,𝑓   𝐵,𝑔   𝑇,𝑓   𝑇,𝑔
Allowed substitution hints:   𝑂(𝑓,𝑔)

Proof of Theorem tendo0cbv
StepHypRef Expression
1 tendo0cbv.o . 2 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
2 eqidd 2621 . . 3 (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
32cbvmptv 4741 . 2 (𝑓𝑇 ↦ ( I ↾ 𝐵)) = (𝑔𝑇 ↦ ( I ↾ 𝐵))
41, 3eqtri 2642 1 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ↦ cmpt 4720   I cid 5013   ↾ cres 5106 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-mpt 4721 This theorem is referenced by:  tendo02  35894  tendo0cl  35897
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