Theorem afvres 44551

Description: The value of a restricted function, analogous to fvres 6775. (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 227	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3		dmres 5902	. . . . . . . 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 412	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
6		snssi 4738	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
7	6	resabs1d 5911	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
8	7	eqcomd 2744	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
9	8	funeqd 6440	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
10	9	biimpd 228	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
11	5, 10	anim12d 608	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
12	11	impcom 407	. . . 4 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
13		df-dfat 44498	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
14		afvfundmfveq 44517	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
15	13, 14	sylbir 234	. . . 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 6775	. . . 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 44498	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20		afvfundmfveq 44517	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
21	19, 20	sylbir 234	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹‘𝐴))
22	21	eqcomd 2744	. . . 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 2782	. 2 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
25		pm3.4 806	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
26	1, 25	sylbi 216	. . . . . . . . 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 6440	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
30	29	biimpd 228	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
31	28, 30	anim12d 608	. . . . . 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 407	. . . 4 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
34		afvnfundmuv 44518	. . . . 5 ⊢ (¬ (𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = V)
35	13, 34	sylnbir 330	. . . 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 44518	. . . . . 6 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
38	19, 37	sylnbir 330	. . . . 5 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
39	38	eqcomd 2744	. . . 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 2778	. 2 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
42	24, 41	pm2.61ian 808	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882  {csn 4558  dom cdm 5580   ↾ cres 5582  Fun wfun 6412  ‘cfv 6418   defAt wdfat 44495  '''cafv 44496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-fv 6426  df-aiota 44464  df-dfat 44498  df-afv 44499

This theorem is referenced by: (None)