Theorem tendoicbv 40775

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 5888	. . . . 5 ⊢ (𝑠 = 𝑢 → ◡(𝑠‘𝑓) = ◡(𝑢‘𝑓))
4	3	mpteq2dv 5249	. . . 4 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)))
5		fveq2 6906	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
6	5	cnveqd 5888	. . . . 5 ⊢ (𝑓 = 𝑔 → ◡(𝑢‘𝑓) = ◡(𝑢‘𝑔))
7	6	cbvmptv 5260	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))
8	4, 7	eqtrdi 2790	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
9	8	cbvmptv 5260	. 2 ⊢ (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
10	1, 9	eqtri 2762	1 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))

Syntax hints:   = wceq 1536   ↦ cmpt 5230  ^◡ccnv 5687  ‘cfv 6562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-cnv 5696  df-iota 6515  df-fv 6570

This theorem is referenced by:  tendoi  40776