Theorem tendoi 38735

Description: Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)

Hypotheses

Ref	Expression
tendoi.i	⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)))
tendoi.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)

Assertion

Ref	Expression
tendoi	⊢ (𝑆 ∈ 𝐸 → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ◡(𝑆‘𝑔)))

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.

Step	Hyp	Ref	Expression
1		fveq1 6755	. . . 4 ⊢ (𝑢 = 𝑆 → (𝑢‘𝑔) = (𝑆‘𝑔))
2	1	cnveqd 5773	. . 3 ⊢ (𝑢 = 𝑆 → ◡(𝑢‘𝑔) = ◡(𝑆‘𝑔))
3	2	mpteq2dv 5172	. 2 ⊢ (𝑢 = 𝑆 → (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)) = (𝑔 ∈ 𝑇 ↦ ◡(𝑆‘𝑔)))
4		tendoi.i	. . 3 ⊢ 𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ◡(𝑠‘𝑓)))
5	4	tendoicbv 38734	. 2 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
6		tendoi.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7	3, 5, 6	mptfvmpt 7086	1 ⊢ (𝑆 ∈ 𝐸 → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ◡(𝑆‘𝑔)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108   ↦ cmpt 5153  ^◡ccnv 5579  ‘cfv 6418  LTrncltrn 38042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426

This theorem is referenced by:  tendoi2  38736  tendoicl  38737