Theorem tendo02 37950

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 2821	. 2 ⊢ (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2		tendo0cbv.o	. . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
3	2	tendo0cbv 37949	. 2 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4		funi 6368	. . 3 ⊢ Fun I
5		tendo02.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6	5	fvexi 6665	. . 3 ⊢ 𝐵 ∈ V
7		resfunexg 6959	. . 3 ⊢ ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
8	4, 6, 7	mp2an 690	. 2 ⊢ ( I ↾ 𝐵) ∈ V
9	1, 3, 8	fvmpt 6749	1 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3481   ↦ cmpt 5127   I cid 5440   ↾ cres 5538  Fun wfun 6330  ‘cfv 6336  Basecbs 16461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pr 5311

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3012  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3483  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4820  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-id 5441  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 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344

This theorem is referenced by:  tendo0co2  37951  tendo0tp  37952  tendo0pl  37954  tendoipl  37960  tendoid0  37988  tendo0mul  37989  tendo0mulr  37990  tendo1ne0  37991  tendoex  38138  dicn0  38355  dihordlem7b  38378  dihmeetlem1N  38453  dihglblem5apreN  38454  dihmeetlem4preN  38469  dihmeetlem13N  38482