Theorem tendo02 38827

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 2734	. 2 ⊢ (𝑔 = 𝐹 → ( I ↾ 𝐵) = ( I ↾ 𝐵))
2		tendo0cbv.o	. . 3 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
3	2	tendo0cbv 38826	. 2 ⊢ 𝑂 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵))
4		funi 6483	. . 3 ⊢ Fun I
5		tendo02.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
6	5	fvexi 6806	. . 3 ⊢ 𝐵 ∈ V
7		resfunexg 7111	. . 3 ⊢ ((Fun I ∧ 𝐵 ∈ V) → ( I ↾ 𝐵) ∈ V)
8	4, 6, 7	mp2an 688	. 2 ⊢ ( I ↾ 𝐵) ∈ V
9	1, 3, 8	fvmpt 6895	1 ⊢ (𝐹 ∈ 𝑇 → (𝑂‘𝐹) = ( I ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2101  Vcvv 3434   ↦ cmpt 5160   I cid 5490   ↾ cres 5593  Fun wfun 6441  ‘cfv 6447  Basecbs 16940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455

This theorem is referenced by:  tendo0co2  38828  tendo0tp  38829  tendo0pl  38831  tendoipl  38837  tendoid0  38865  tendo0mul  38866  tendo0mulr  38867  tendo1ne0  38868  tendoex  39015  dicn0  39232  dihordlem7b  39255  dihmeetlem1N  39330  dihglblem5apreN  39331  dihmeetlem4preN  39346  dihmeetlem13N  39359