Theorem tendoicbv 40794

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 6860	. . . . . 6 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓))
3	2	cnveqd 5842	. . . . 5 ⊢ (𝑠 = 𝑢 → ◡(𝑠‘𝑓) = ◡(𝑢‘𝑓))
4	3	mpteq2dv 5204	. . . 4 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)))
5		fveq2 6861	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
6	5	cnveqd 5842	. . . . 5 ⊢ (𝑓 = 𝑔 → ◡(𝑢‘𝑓) = ◡(𝑢‘𝑔))
7	6	cbvmptv 5214	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))
8	4, 7	eqtrdi 2781	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
9	8	cbvmptv 5214	. 2 ⊢ (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
10	1, 9	eqtri 2753	1 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))

Syntax hints:   = wceq 1540   ↦ cmpt 5191  ^◡ccnv 5640  ‘cfv 6514

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-cnv 5649  df-iota 6467  df-fv 6522

This theorem is referenced by:  tendoi  40795