Theorem tendo02 41279

Description: Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
tendo0cbv.o	⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
tendo02.b	⊢ 𝐵 = (Base‘𝐾)

Assertion

Ref	Expression
tendo02	⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))

Distinct variable groups:   𝐵,𝑓   𝑇,𝑓

Allowed substitution hints:   𝐹(𝑓)   𝐾(𝑓)   𝑂(𝑓)

Proof of Theorem tendo02

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2740	. 2 ⊢ (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2		tendo0cbv.o	. . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
3	2	tendo0cbv 41278	. 2 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4		funi 6517	. . 3 ⊢ Fun I
5		tendo02.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6	5	fvexi 6841	. . 3 ⊢ 𝐵 ∈ V
7		resfunexg 7159	. . 3 ⊢ ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
8	4, 6, 7	mp2an 698	. 2 ⊢ ( I ↾ 𝐵) ∈ V
9	1, 3, 8	fvmpt 6935	1 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ↦ cmpt 5153   I cid 5512   ↾ cres 5620  Fun wfun 6479  ‘cfv 6485  Basecbs 17170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493

This theorem is referenced by:  tendo0co2  41280  tendo0tp  41281  tendo0pl  41283  tendoipl  41289  tendoid0  41317  tendo0mul  41318  tendo0mulr  41319  tendo1ne0  41320  tendoex  41467  dicn0  41684  dihordlem7b  41707  dihmeetlem1N  41782  dihglblem5apreN  41783  dihmeetlem4preN  41798  dihmeetlem13N  41811