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Theorem afvres 41766
Description: The value of a restricted function, analogous to fvres 6348. (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 3945 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴𝐵𝐴 ∈ dom 𝐹))
21biimpri 218 . . . . . . . 8 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3 dmres 5560 . . . . . . . 8 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
42, 3syl6eleqr 2860 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹𝐵))
54ex 397 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹𝐵)))
6 snssi 4472 . . . . . . . . . 10 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
76resabs1d 5569 . . . . . . . . 9 (𝐴𝐵 → ((𝐹𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
87eqcomd 2776 . . . . . . . 8 (𝐴𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹𝐵) ↾ {𝐴}))
98funeqd 6053 . . . . . . 7 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹𝐵) ↾ {𝐴})))
109biimpd 219 . . . . . 6 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹𝐵) ↾ {𝐴})))
115, 10anim12d 588 . . . . 5 (𝐴𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
1211impcom 394 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
13 df-dfat 41710 . . . . 5 ((𝐹𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
14 afvfundmfveq 41732 . . . . 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 6348 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
1817adantl 467 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
19 df-dfat 41710 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20 afvfundmfveq 41732 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
2119, 20sylbir 225 . . . . 5 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹𝐴))
2221eqcomd 2776 . . . 4 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) = (𝐹'''𝐴))
2322adantr 466 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐹𝐴) = (𝐹'''𝐴))
2416, 18, 233eqtrd 2808 . 2 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
25 pm3.4 811 . . . . . . . . . 10 ((𝐴𝐵𝐴 ∈ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
261, 25sylbi 207 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2726, 3eleq2s 2867 . . . . . . . 8 (𝐴 ∈ dom (𝐹𝐵) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2827com12 32 . . . . . . 7 (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) → 𝐴 ∈ dom 𝐹))
297funeqd 6053 . . . . . . . 8 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
3029biimpd 219 . . . . . . 7 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
3128, 30anim12d 588 . . . . . 6 (𝐴𝐵 → ((𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
3231con3d 149 . . . . 5 (𝐴𝐵 → (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
3332impcom 394 . . . 4 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
34 afvnfundmuv 41733 . . . . 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 41733 . . . . . 6 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
3819, 37sylnbir 320 . . . . 5 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
3938eqcomd 2776 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → V = (𝐹'''𝐴))
4039adantr 466 . . 3 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → V = (𝐹'''𝐴))
4136, 40eqtrd 2804 . 2 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
4224, 41pm2.61ian 795 1 (𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  cin 3720  {csn 4314  dom cdm 5249  cres 5251  Fun wfun 6025  cfv 6031   defAt wdfat 41707  '''cafv 41708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-dfat 41710  df-afv 41711
This theorem is referenced by: (None)
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