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Theorem afvres 41573
Description: The value of a restricted function, analogous to fvres 6245. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
Assertion
Ref Expression
afvres (𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))

Proof of Theorem afvres
StepHypRef Expression
1 elin 3829 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴𝐵𝐴 ∈ dom 𝐹))
21biimpri 218 . . . . . . . 8 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3 dmres 5454 . . . . . . . 8 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
42, 3syl6eleqr 2741 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹𝐵))
54ex 449 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹𝐵)))
6 snssi 4371 . . . . . . . . . 10 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
76resabs1d 5463 . . . . . . . . 9 (𝐴𝐵 → ((𝐹𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
87eqcomd 2657 . . . . . . . 8 (𝐴𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹𝐵) ↾ {𝐴}))
98funeqd 5948 . . . . . . 7 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹𝐵) ↾ {𝐴})))
109biimpd 219 . . . . . 6 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹𝐵) ↾ {𝐴})))
115, 10anim12d 585 . . . . 5 (𝐴𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
1211impcom 445 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
13 df-dfat 41517 . . . . 5 ((𝐹𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
14 afvfundmfveq 41539 . . . . 5 ((𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)'''𝐴) = ((𝐹𝐵)‘𝐴))
1513, 14sylbir 225 . . . 4 ((𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})) → ((𝐹𝐵)'''𝐴) = ((𝐹𝐵)‘𝐴))
1612, 15syl 17 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = ((𝐹𝐵)‘𝐴))
17 fvres 6245 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
1817adantl 481 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
19 df-dfat 41517 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20 afvfundmfveq 41539 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
2119, 20sylbir 225 . . . . 5 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹𝐴))
2221eqcomd 2657 . . . 4 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) = (𝐹'''𝐴))
2322adantr 480 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐹𝐴) = (𝐹'''𝐴))
2416, 18, 233eqtrd 2689 . 2 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
25 pm3.4 583 . . . . . . . . . 10 ((𝐴𝐵𝐴 ∈ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
261, 25sylbi 207 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2726, 3eleq2s 2748 . . . . . . . 8 (𝐴 ∈ dom (𝐹𝐵) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2827com12 32 . . . . . . 7 (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) → 𝐴 ∈ dom 𝐹))
297funeqd 5948 . . . . . . . 8 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
3029biimpd 219 . . . . . . 7 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
3128, 30anim12d 585 . . . . . 6 (𝐴𝐵 → ((𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
3231con3d 148 . . . . 5 (𝐴𝐵 → (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
3332impcom 445 . . . 4 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
34 afvnfundmuv 41540 . . . . 5 (¬ (𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)'''𝐴) = V)
3513, 34sylnbir 320 . . . 4 (¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})) → ((𝐹𝐵)'''𝐴) = V)
3633, 35syl 17 . . 3 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = V)
37 afvnfundmuv 41540 . . . . . 6 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
3819, 37sylnbir 320 . . . . 5 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
3938eqcomd 2657 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → V = (𝐹'''𝐴))
4039adantr 480 . . 3 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → V = (𝐹'''𝐴))
4136, 40eqtrd 2685 . 2 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
4224, 41pm2.61ian 848 1 (𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  {csn 4210  dom cdm 5143  cres 5145  Fun wfun 5920  cfv 5926   defAt wdfat 41514  '''cafv 41515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-dfat 41517  df-afv 41518
This theorem is referenced by: (None)
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