Theorem tendo0cbv 39960

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 2731	. . 3 ⊢ (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
3	2	cbvmptv 5260	. 2 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4	1, 3	eqtri 2758	1 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))

Syntax hints:   = wceq 1539   ↦ cmpt 5230   I cid 5572   ↾ cres 5677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-opab 5210  df-mpt 5231

This theorem is referenced by:  tendo02  39961  tendo0cl  39964