Theorem afvres 48241

Description: The value of a restricted function, analogous to fvres 6904. (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 3915	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹))
2	1	biimpri 231	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3		dmres 6003	. . . . . . . 8 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
4	2, 3	eleqtrrdi 2872	. . . . . . 7 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹 ↾ 𝐵))
5	4	ex 418	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
6		snssi 4746	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
7	6	resabs1d 5999	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
8	7	eqcomd 2767	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
9	8	funeqd 6561	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
10	9	biimpd 232	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
11	5, 10	anim12d 621	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
12	11	impcom 413	. . . 4 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
13		df-dfat 48188	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
14		afvfundmfveq 48207	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
15	13, 14	sylbir 238	. . . 4 ⊢ ((𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
16	12, 15	syl 18	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = ((𝐹 ↾ 𝐵)‘𝐴))
17		fvres 6904	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
18	17	adantl 487	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)‘𝐴) = (𝐹‘𝐴))
19		df-dfat 48188	. . . . . 6 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20		afvfundmfveq 48207	. . . . . 6 ⊢ (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹‘𝐴))
21	19, 20	sylbir 238	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹‘𝐴))
22	21	eqcomd 2767	. . . 4 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹‘𝐴) = (𝐹'''𝐴))
23	22	adantr 486	. . 3 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → (𝐹‘𝐴) = (𝐹'''𝐴))
24	16, 18, 23	3eqtrd 2800	. 2 ⊢ (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
25		pm3.4 822	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
26	1, 25	sylbi 220	. . . . . . . . 9 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
27	26, 3	eleq2s 2879	. . . . . . . 8 ⊢ (𝐴 ∈ dom (𝐹 ↾ 𝐵) → (𝐴 ∈ 𝐵 → 𝐴 ∈ dom 𝐹))
28	27	com12 33	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom (𝐹 ↾ 𝐵) → 𝐴 ∈ dom 𝐹))
29	7	funeqd 6561	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
30	29	biimpd 232	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
31	28, 30	anim12d 621	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
32	31	con3d 153	. . . . 5 ⊢ (𝐴 ∈ 𝐵 → (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → ¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
33	32	impcom 413	. . . 4 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
34		afvnfundmuv 48208	. . . . 5 ⊢ (¬ (𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)'''𝐴) = V)
35	13, 34	sylnbir 334	. . . 4 ⊢ (¬ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → ((𝐹 ↾ 𝐵)'''𝐴) = V)
36	33, 35	syl 18	. . 3 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = V)
37		afvnfundmuv 48208	. . . . . 6 ⊢ (¬ 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
38	19, 37	sylnbir 334	. . . . 5 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
39	38	eqcomd 2767	. . . 4 ⊢ (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → V = (𝐹'''𝐴))
40	39	adantr 486	. . 3 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → V = (𝐹'''𝐴))
41	36, 40	eqtrd 2796	. 2 ⊢ ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))
42	24, 41	pm2.61ian 824	1 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵)'''𝐴) = (𝐹'''𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∩ cin 3898  {csn 4584  dom cdm 5651   ↾ cres 5653  Fun wfun 6532  ‘cfv 6538   defAt wdfat 48185  '''cafv 48186

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-aiota 48154  df-dfat 48188  df-afv 48189

This theorem is used by: (None)