Theorem tendo0cbv 41560

Description: Define additive identity for trace-preserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 11-Jun-2013.)

Hypothesis

Ref	Expression
tendo0cbv.o	⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))

Assertion

Ref	Expression
tendo0cbv	⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))

Distinct variable groups:   𝐵,𝑓,𝑔   𝑇,𝑓,𝑔

Allowed substitution hints:   𝑂(𝑓,𝑔)

Proof of Theorem tendo0cbv

Step	Hyp	Ref	Expression
1		tendo0cbv.o	. 2 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
2		eqidd 2764	. . 3 ⊢ (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
3	2	cbvmptv 5215	. 2 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4	1, 3	eqtri 2786	1 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))

Syntax hints:   = wceq 1570   ↦ cmpt 5192   I cid 5555   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-opab 5174  df-mpt 5193

This theorem is referenced by:  tendo02  41561  tendo0cl  41564