Theorem tendo02 41987

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 2764	. 2 ⊢ (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2		tendo0cbv.o	. . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
3	2	tendo0cbv 41986	. 2 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4		funi 6581	. . 3 ⊢ Fun I
5		tendo02.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6	5	fvexi 6910	. . 3 ⊢ 𝐵 ∈ V
7		resfunexg 7231	. . 3 ⊢ ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
8	4, 6, 7	mp2an 705	. 2 ⊢ ( I ↾ 𝐵) ∈ V
9	1, 3, 8	fvmpt 7004	1 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2146  Vcvv 3454   ↦ cmpt 5189   I cid 5549   ↾ cres 5657  Fun wfun 6541  ‘cfv 6547  Basecbs 17402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555

This theorem is used by:  tendo0co2  41988  tendo0tp  41989  tendo0pl  41991  tendoipl  41997  tendoid0  42025  tendo0mul  42026  tendo0mulr  42027  tendo1ne0  42028  tendoex  42175  dicn0  42392  dihordlem7b  42415  dihmeetlem1N  42490  dihglblem5apreN  42491  dihmeetlem4preN  42506  dihmeetlem13N  42519