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Theorem tendo02 35541
Description: Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendo0cbv.o 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
tendo02.b 𝐵 = (Base‘𝐾)
Assertion
Ref Expression
tendo02 (𝐹𝑇 → (𝑂𝐹) = ( 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.
StepHypRef Expression
1 eqidd 2627 . 2 (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2 tendo0cbv.o . . 3 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
32tendo0cbv 35540 . 2 𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))
4 funi 5880 . . 3 Fun I
5 tendo02.b . . . 4 𝐵 = (Base‘𝐾)
6 fvex 6160 . . . 4 (Base‘𝐾) ∈ V
75, 6eqeltri 2700 . . 3 𝐵 ∈ V
8 resfunexg 6434 . . 3 ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
94, 7, 8mp2an 707 . 2 ( I ↾ 𝐵) ∈ V
101, 3, 9fvmpt 6240 1 (𝐹𝑇 → (𝑂𝐹) = ( I ↾ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  Vcvv 3191  cmpt 4678   I cid 4989  cres 5081  Fun wfun 5844  cfv 5850  Basecbs 15776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858
This theorem is referenced by:  tendo0co2  35542  tendo0tp  35543  tendo0pl  35545  tendoipl  35551  tendoid0  35579  tendo0mul  35580  tendo0mulr  35581  tendo1ne0  35582  tendoex  35729  dicn0  35947  dihordlem7b  35970  dihmeetlem1N  36045  dihglblem5apreN  36046  dihmeetlem4preN  36061  dihmeetlem13N  36074
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