Theorem afvres 47635

Description: The value of a restricted function, analogous to fvres 6846. (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 3899	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹))
2	1	biimpri 229	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3		dmres 5964	. . . . . . . 8 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
4	2, 3	eleqtrrdi 2850	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹 ↾ 𝐵))
5	4	ex 413	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
6		snssi 4717	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
7	6	resabs1d 5960	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
8	7	eqcomd 2745	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
9	8	funeqd 6507	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
10	9	biimpd 230	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
11	5, 10	anim12d 615	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
12	11	impcom 408	. . . 4 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
13		df-dfat 47582	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
14		afvfundmfveq 47601	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
15	13, 14	sylbir 236	. . . 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 6846	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
18	17	adantl 482	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
19		df-dfat 47582	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20		afvfundmfveq 47601	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
21	19, 20	sylbir 236	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹‘𝐴))
22	21	eqcomd 2745	. . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹‘𝐴) = (𝐹'''𝐴))
23	22	adantr 481	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = (𝐹'''𝐴))
24	16, 18, 23	3eqtrd 2778	. 2 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
25		pm3.4 815	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
26	1, 25	sylbi 218	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
27	26, 3	eleq2s 2857	. . . . . . . 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 6507	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
30	29	biimpd 230	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
31	28, 30	anim12d 615	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
32	31	con3d 152	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → ¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
33	32	impcom 408	. . . 4 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
34		afvnfundmuv 47602	. . . . 5 ⊢ (¬ (𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = V)
35	13, 34	sylnbir 332	. . . 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 47602	. . . . . 6 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
38	19, 37	sylnbir 332	. . . . 5 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
39	38	eqcomd 2745	. . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → V = (𝐹'''𝐴))
40	39	adantr 481	. . 3 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → V = (𝐹'''𝐴))
41	36, 40	eqtrd 2774	. 2 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
42	24, 41	pm2.61ian 817	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∩ cin 3882  {csn 4555  dom cdm 5618   ↾ cres 5620  Fun wfun 6479  ‘cfv 6485   defAt wdfat 47579  '''cafv 47580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-res 5630  df-iota 6441  df-fun 6487  df-fv 6493  df-aiota 47548  df-dfat 47582  df-afv 47583

This theorem is referenced by: (None)