Theorem tendoicbv 36081

Description: Define inverse function for trace-perserving 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 6190	. . . . . 6 ⊢ (𝑠 = 𝑢 → (𝑠‘𝑓) = (𝑢‘𝑓))
3	2	cnveqd 5298	. . . . 5 ⊢ (𝑠 = 𝑢 → ◡(𝑠‘𝑓) = ◡(𝑢‘𝑓))
4	3	mpteq2dv 4745	. . . 4 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)))
5		fveq2 6191	. . . . . 6 ⊢ (𝑓 = 𝑔 → (𝑢‘𝑓) = (𝑢‘𝑔))
6	5	cnveqd 5298	. . . . 5 ⊢ (𝑓 = 𝑔 → ◡(𝑢‘𝑓) = ◡(𝑢‘𝑔))
7	6	cbvmptv 4750	. . . 4 ⊢ (𝑓 ∈ 𝑇 ↦ ◡(𝑢‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔))
8	4, 7	syl6eq 2672	. . 3 ⊢ (𝑠 = 𝑢 → (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
9	8	cbvmptv 4750	. 2 ⊢ (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓))) = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
10	1, 9	eqtri 2644	1 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))

Syntax hints:   = wceq 1483   ↦ cmpt 4729  ^◡ccnv 5113  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-iota 5851  df-fv 5896

This theorem is referenced by:  tendoi  36082