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Theorem tendoi 39260
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 6842 . . . 4 (𝑢 = 𝑆 → (𝑢𝑔) = (𝑆𝑔))
21cnveqd 5832 . . 3 (𝑢 = 𝑆(𝑢𝑔) = (𝑆𝑔))
32mpteq2dv 5208 . 2 (𝑢 = 𝑆 → (𝑔𝑇(𝑢𝑔)) = (𝑔𝑇(𝑆𝑔)))
4 tendoi.i . . 3 𝐼 = (𝑠𝐸 ↦ (𝑓𝑇(𝑠𝑓)))
54tendoicbv 39259 . 2 𝐼 = (𝑢𝐸 ↦ (𝑔𝑇(𝑢𝑔)))
6 tendoi.t . 2 𝑇 = ((LTrn‘𝐾)‘𝑊)
73, 5, 6mptfvmpt 7179 1 (𝑆𝐸 → (𝐼𝑆) = (𝑔𝑇(𝑆𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cmpt 5189  ccnv 5633  cfv 6497  LTrncltrn 38567
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  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-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  tendoi2  39261  tendoicl  39262
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