Theorem afvres 39901

Description: The value of a restricted function, analogous to fvres 6117. (Contributed by Alexander van der Vekens, 22-Jul-2017.)

Assertion

Ref	Expression
afvres	⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))

Proof of Theorem afvres

Step	Hyp	Ref	Expression
1		elin 3758	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹))
2	1	biimpri 217	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3		dmres 5339	. . . . . . . 8 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
4	2, 3	syl6eleqr 2699	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹 ↾ 𝐵))
5	4	ex 449	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
6		snssi 4280	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
7	6	resabs1d 5348	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
8	7	eqcomd 2616	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
9	8	funeqd 5825	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
10	9	biimpd 218	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
11	5, 10	anim12d 584	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
12	11	impcom 445	. . . 4 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
13		df-dfat 39845	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
14		afvfundmfveq 39867	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
15	13, 14	sylbir 224	. . . 4 ⊢ ((𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
16	12, 15	syl 17	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
17		fvres 6117	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
18	17	adantl 481	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
19		df-dfat 39845	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20		afvfundmfveq 39867	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
21	19, 20	sylbir 224	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹‘𝐴))
22	21	eqcomd 2616	. . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹‘𝐴) = (𝐹'''𝐴))
23	22	adantr 480	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = (𝐹'''𝐴))
24	16, 18, 23	3eqtrd 2648	. 2 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
25		pm3.4 582	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
26	1, 25	sylbi 206	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
27	26, 3	eleq2s 2706	. . . . . . . 8 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
28	27	com12 32	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ dom 𝐹))
29	7	funeqd 5825	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
30	29	biimpd 218	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
31	28, 30	anim12d 584	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
32	31	con3d 147	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → ¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
33	32	impcom 445	. . . 4 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
34		afvnfundmuv 39868	. . . . 5 ⊢ (¬ (𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = V)
35	13, 34	sylnbir 320	. . . 4 ⊢ (¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → ((𝐹 ↾ 𝐵)'''𝐴) = V)
36	33, 35	syl 17	. . 3 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = V)
37		afvnfundmuv 39868	. . . . . 6 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
38	19, 37	sylnbir 320	. . . . 5 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
39	38	eqcomd 2616	. . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → V = (𝐹'''𝐴))
40	39	adantr 480	. . 3 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → V = (𝐹'''𝐴))
41	36, 40	eqtrd 2644	. 2 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
42	24, 41	pm2.61ian 827	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539  {csn 4125  dom cdm 5038   ↾ cres 5040  Fun wfun 5798  ‘cfv 5804   defAt wdfat 39842  '''cafv 39843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-dfat 39845  df-afv 39846

This theorem is referenced by: (None)