Theorem tendoi 38494

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 6694	. . . 4 ⊢ (𝑢 = 𝑆 → (𝑢‘𝑔) = (𝑆‘𝑔))
2	1	cnveqd 5729	. . 3 ⊢ (𝑢 = 𝑆 → ◡(𝑢‘𝑔) = ◡(𝑆‘𝑔))
3	2	mpteq2dv 5136	. 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 38493	. 2 ⊢ 𝐼 = (𝑢 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ◡(𝑢‘𝑔)))
6		tendoi.t	. 2 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
7	3, 5, 6	mptfvmpt 7022	1 ⊢ (𝑆 ∈ 𝐸 → (𝐼‘𝑆) = (𝑔 ∈ 𝑇 ↦ ◡(𝑆‘𝑔)))

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2112   ↦ cmpt 5120  ^◡ccnv 5535  ‘cfv 6358  LTrncltrn 37801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pr 5307

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366

This theorem is referenced by:  tendoi2  38495  tendoicl  38496