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Theorem tendoicbv 40177
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 6884 . . . . . 6 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32cnveqd 5869 . . . . 5 (𝑠 = 𝑢(𝑠𝑓) = (𝑢𝑓))
43mpteq2dv 5243 . . . 4 (𝑠 = 𝑢 → (𝑓𝑇(𝑠𝑓)) = (𝑓𝑇(𝑢𝑓)))
5 fveq2 6885 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
65cnveqd 5869 . . . . 5 (𝑓 = 𝑔(𝑢𝑓) = (𝑢𝑔))
76cbvmptv 5254 . . . 4 (𝑓𝑇(𝑢𝑓)) = (𝑔𝑇(𝑢𝑔))
84, 7eqtrdi 2782 . . 3 (𝑠 = 𝑢 → (𝑓𝑇(𝑠𝑓)) = (𝑔𝑇(𝑢𝑔)))
98cbvmptv 5254 . 2 (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓))) = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
101, 9eqtri 2754 1 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cmpt 5224  ccnv 5668  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-cnv 5677  df-iota 6489  df-fv 6545
This theorem is referenced by:  tendoi  40178
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