Theorem tendo0cbv 39322

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 2732	. . 3 ⊢ (𝑓 = 𝑔 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
3	2	cbvmptv 5223	. 2 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4	1, 3	eqtri 2759	1 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))

Syntax hints:   = wceq 1541   ↦ cmpt 5193   I cid 5535   ↾ cres 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-mpt 5194

This theorem is referenced by:  tendo02  39323  tendo0cl  39326