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Theorem tendoi 36603
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 6331 . . . 4 (𝑢 = 𝑆 → (𝑢𝑔) = (𝑆𝑔))
21cnveqd 5436 . . 3 (𝑢 = 𝑆(𝑢𝑔) = (𝑆𝑔))
32mpteq2dv 4879 . 2 (𝑢 = 𝑆 → (𝑔𝑇(𝑢𝑔)) = (𝑔𝑇(𝑆𝑔)))
4 tendoi.i . . 3 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
54tendoicbv 36602 . 2 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
6 tendoi.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
7 fvex 6342 . . . 4 ((LTrn‘𝐾)‘𝑊) ∈ V
86, 7eqeltri 2846 . . 3 𝑇 ∈ V
98mptex 6630 . 2 (𝑔𝑇(𝑆𝑔)) ∈ V
103, 5, 9fvmpt 6424 1 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  cmpt 4863  ccnv 5248  cfv 6031  LTrncltrn 35909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  tendoi2  36604  tendoicl  36605
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