Theorem tendoicbv 40795

Description: Define inverse function for trace-preserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)

Hypothesis

Ref	Expression
tendoi.i	⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)))

Assertion

Ref	Expression
tendoicbv	⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))

Distinct variable groups:   𝑢,𝑠,𝐸   𝑓,𝑔,𝑠,𝑢,𝑇

Allowed substitution hints:   𝐸(𝑓,𝑔)   𝐼(𝑢,𝑓,𝑔,𝑠)

Proof of Theorem tendoicbv

Step	Hyp	Ref	Expression
1		tendoi.i	. 2 ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)))
2		fveq1 6905	. . . . . 6 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓))
3	2	cnveqd 5886	. . . . 5 ⊢ (𝑠 = 𝑢 → ◡(𝑠‘𝑓) = ◡(𝑢‘𝑓))
4	3	mpteq2dv 5244	. . . 4 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)))
5		fveq2 6906	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
6	5	cnveqd 5886	. . . . 5 ⊢ (𝑓 = 𝑔 → ◡(𝑢‘𝑓) = ◡(𝑢‘𝑔))
7	6	cbvmptv 5255	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))
8	4, 7	eqtrdi 2793	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
9	8	cbvmptv 5255	. 2 ⊢ (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
10	1, 9	eqtri 2765	1 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))

Syntax hints:   = wceq 1540   ↦ cmpt 5225  ^◡ccnv 5684  ‘cfv 6561

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-cnv 5693  df-iota 6514  df-fv 6569

This theorem is referenced by:  tendoi  40796