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Theorem tendoicbv 39259
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 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
Assertion
Ref Expression
tendoicbv 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
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
StepHypRef Expression
1 tendoi.i . 2 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
2 fveq1 6842 . . . . . 6 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32cnveqd 5832 . . . . 5 (𝑠 = 𝑢(𝑠𝑓) = (𝑢𝑓))
43mpteq2dv 5208 . . . 4 (𝑠 = 𝑢 → (𝑓𝑇(𝑠𝑓)) = (𝑓𝑇(𝑢𝑓)))
5 fveq2 6843 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
65cnveqd 5832 . . . . 5 (𝑓 = 𝑔(𝑢𝑓) = (𝑢𝑔))
76cbvmptv 5219 . . . 4 (𝑓𝑇(𝑢𝑓)) = (𝑔𝑇(𝑢𝑔))
84, 7eqtrdi 2793 . . 3 (𝑠 = 𝑢 → (𝑓𝑇(𝑠𝑓)) = (𝑔𝑇(𝑢𝑔)))
98cbvmptv 5219 . 2 (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓))) = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
101, 9eqtri 2765 1 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cmpt 5189  ccnv 5633  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-cnv 5642  df-iota 6449  df-fv 6505
This theorem is referenced by:  tendoi  39260
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