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Theorem afvres 47122
Description: The value of a restricted function, analogous to fvres 6926. (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 3979 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴𝐵𝐴 ∈ dom 𝐹))
21biimpri 228 . . . . . . . 8 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3 dmres 6032 . . . . . . . 8 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
42, 3eleqtrrdi 2850 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹𝐵))
54ex 412 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹𝐵)))
6 snssi 4813 . . . . . . . . . 10 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
76resabs1d 6028 . . . . . . . . 9 (𝐴𝐵 → ((𝐹𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
87eqcomd 2741 . . . . . . . 8 (𝐴𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹𝐵) ↾ {𝐴}))
98funeqd 6590 . . . . . . 7 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹𝐵) ↾ {𝐴})))
109biimpd 229 . . . . . 6 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹𝐵) ↾ {𝐴})))
115, 10anim12d 609 . . . . 5 (𝐴𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
1211impcom 407 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
13 df-dfat 47069 . . . . 5 ((𝐹𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
14 afvfundmfveq 47088 . . . . 5 ((𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)'''𝐴) = ((𝐹𝐵)‘𝐴))
1513, 14sylbir 235 . . . 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 6926 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
1817adantl 481 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
19 df-dfat 47069 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20 afvfundmfveq 47088 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
2119, 20sylbir 235 . . . . 5 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹𝐴))
2221eqcomd 2741 . . . 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 2779 . 2 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
25 pm3.4 810 . . . . . . . . . 10 ((𝐴𝐵𝐴 ∈ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
261, 25sylbi 217 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2726, 3eleq2s 2857 . . . . . . . 8 (𝐴 ∈ dom (𝐹𝐵) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2827com12 32 . . . . . . 7 (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) → 𝐴 ∈ dom 𝐹))
297funeqd 6590 . . . . . . . 8 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
3029biimpd 229 . . . . . . 7 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
3128, 30anim12d 609 . . . . . 6 (𝐴𝐵 → ((𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
3231con3d 152 . . . . 5 (𝐴𝐵 → (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
3332impcom 407 . . . 4 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
34 afvnfundmuv 47089 . . . . 5 (¬ (𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)'''𝐴) = V)
3513, 34sylnbir 331 . . . 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 47089 . . . . . 6 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
3819, 37sylnbir 331 . . . . 5 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
3938eqcomd 2741 . . . 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 2775 . 2 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
4224, 41pm2.61ian 812 1 (𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  {csn 4631  dom cdm 5689  cres 5691  Fun wfun 6557  cfv 6563   defAt wdfat 47066  '''cafv 47067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-aiota 47035  df-dfat 47069  df-afv 47070
This theorem is referenced by: (None)
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