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Theorem afvres 40543
Description: The value of a restricted function, analogous to fvres 6165. (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 3779 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴𝐵𝐴 ∈ dom 𝐹))
21biimpri 218 . . . . . . . 8 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3 dmres 5382 . . . . . . . 8 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
42, 3syl6eleqr 2715 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹𝐵))
54ex 450 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹𝐵)))
6 snssi 4313 . . . . . . . . . 10 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
76resabs1d 5391 . . . . . . . . 9 (𝐴𝐵 → ((𝐹𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
87eqcomd 2632 . . . . . . . 8 (𝐴𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹𝐵) ↾ {𝐴}))
98funeqd 5871 . . . . . . 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 446 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
13 df-dfat 40487 . . . . 5 ((𝐹𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
14 afvfundmfveq 40509 . . . . 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 6165 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
1817adantl 482 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
19 df-dfat 40487 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20 afvfundmfveq 40509 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
2119, 20sylbir 225 . . . . 5 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹𝐴))
2221eqcomd 2632 . . . 4 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) = (𝐹'''𝐴))
2322adantr 481 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐹𝐴) = (𝐹'''𝐴))
2416, 18, 233eqtrd 2664 . 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 2722 . . . . . . . 8 (𝐴 ∈ dom (𝐹𝐵) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2827com12 32 . . . . . . 7 (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) → 𝐴 ∈ dom 𝐹))
297funeqd 5871 . . . . . . . 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 446 . . . 4 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
34 afvnfundmuv 40510 . . . . 5 (¬ (𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)'''𝐴) = V)
3513, 34sylnbir 321 . . . 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 40510 . . . . . 6 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
3819, 37sylnbir 321 . . . . 5 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
3938eqcomd 2632 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → V = (𝐹'''𝐴))
4039adantr 481 . . 3 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → V = (𝐹'''𝐴))
4136, 40eqtrd 2660 . 2 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
4224, 41pm2.61ian 830 1 (𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  cin 3559  {csn 4153  dom cdm 5079  cres 5081  Fun wfun 5844  cfv 5850   defAt wdfat 40484  '''cafv 40485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-iota 5813  df-fun 5852  df-fv 5858  df-dfat 40487  df-afv 40488
This theorem is referenced by: (None)
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