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Theorem tendoicbv 38034
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 6660 . . . . . 6 (𝑠 = 𝑢 → (𝑠𝑓) = (𝑢𝑓))
32cnveqd 5733 . . . . 5 (𝑠 = 𝑢(𝑠𝑓) = (𝑢𝑓))
43mpteq2dv 5148 . . . 4 (𝑠 = 𝑢 → (𝑓𝑇(𝑠𝑓)) = (𝑓𝑇(𝑢𝑓)))
5 fveq2 6661 . . . . . 6 (𝑓 = 𝑔 → (𝑢𝑓) = (𝑢𝑔))
65cnveqd 5733 . . . . 5 (𝑓 = 𝑔(𝑢𝑓) = (𝑢𝑔))
76cbvmptv 5155 . . . 4 (𝑓𝑇(𝑢𝑓)) = (𝑔𝑇(𝑢𝑔))
84, 7syl6eq 2875 . . 3 (𝑠 = 𝑢 → (𝑓𝑇(𝑠𝑓)) = (𝑔𝑇(𝑢𝑔)))
98cbvmptv 5155 . 2 (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓))) = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
101, 9eqtri 2847 1 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cmpt 5132  ccnv 5541  cfv 6343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-v 3482  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-cnv 5550  df-iota 6302  df-fv 6351
This theorem is referenced by:  tendoi  38035
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