Theorem tendo02 41581

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 41580	. 2 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4		funi 6568	. . 3 ⊢ Fun I
5		tendo02.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6	5	fvexi 6895	. . 3 ⊢ 𝐵 ∈ V
7		resfunexg 7213	. . 3 ⊢ ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
8	4, 6, 7	mp2an 704	. 2 ⊢ ( I ↾ 𝐵) ∈ V
9	1, 3, 8	fvmpt 6989	1 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ↦ cmpt 5192   I cid 5555   ↾ cres 5663  Fun wfun 6530  ‘cfv 6536  Basecbs 17264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by:  tendo0co2  41582  tendo0tp  41583  tendo0pl  41585  tendoipl  41591  tendoid0  41619  tendo0mul  41620  tendo0mulr  41621  tendo1ne0  41622  tendoex  41769  dicn0  41986  dihordlem7b  42009  dihmeetlem1N  42084  dihglblem5apreN  42085  dihmeetlem4preN  42100  dihmeetlem13N  42113