Theorem tendoi 40267

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 6896	. . . 4 ⊢ (𝑢 = 𝑆 → (𝑢‘𝑔) = (𝑆‘𝑔))
2	1	cnveqd 5878	. . 3 ⊢ (𝑢 = 𝑆 → ◡(𝑢‘𝑔) = ◡(𝑆‘𝑔))
3	2	mpteq2dv 5250	. 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 40266	. 2 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
6		tendoi.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7	3, 5, 6	mptfvmpt 7240	1 ⊢ (𝑆 ∈ 𝐸 → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ◡(𝑆‘𝑔)))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   ↦ cmpt 5231  ^◡ccnv 5677  ‘cfv 6548  LTrncltrn 39574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556

This theorem is referenced by:  tendoi2  40268  tendoicl  40269