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Theorem tendoi 35100
Description: Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
Hypotheses
Ref Expression
tendoi.i 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
tendoi.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
Assertion
Ref Expression
tendoi (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
Distinct variable groups:   𝐸,𝑠   𝑓,𝑔,𝑠,𝑇   𝑓,𝑊,𝑔,𝑠   𝑆,𝑔
Allowed substitution hints:   𝑆(𝑓,𝑠)   𝐸(𝑓,𝑔)   𝐼(𝑓,𝑔,𝑠)   𝐾(𝑓,𝑔,𝑠)

Proof of Theorem tendoi
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 fveq1 6102 . . . 4 (𝑢 = 𝑆 → (𝑢𝑔) = (𝑆𝑔))
21cnveqd 5220 . . 3 (𝑢 = 𝑆(𝑢𝑔) = (𝑆𝑔))
32mpteq2dv 4673 . 2 (𝑢 = 𝑆 → (𝑔𝑇(𝑢𝑔)) = (𝑔𝑇(𝑆𝑔)))
4 tendoi.i . . 3 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
54tendoicbv 35099 . 2 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
6 tendoi.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
7 fvex 6113 . . . 4 ((LTrn‘𝐾)‘𝑊) ∈ V
86, 7eqeltri 2684 . . 3 𝑇 ∈ V
98mptex 6390 . 2 (𝑔𝑇(𝑆𝑔)) ∈ V
103, 5, 9fvmpt 6191 1 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  cmpt 4643  ccnv 5037  cfv 5804  LTrncltrn 34405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812
This theorem is referenced by:  tendoi2  35101  tendoicl  35102
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