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Theorem afvres 43654
Description: The value of a restricted function, analogous to fvres 6680. (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 3935 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴𝐵𝐴 ∈ dom 𝐹))
21biimpri 231 . . . . . . . 8 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
3 dmres 5862 . . . . . . . 8 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
42, 3eleqtrrdi 2927 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹𝐵))
54ex 416 . . . . . 6 (𝐴𝐵 → (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹𝐵)))
6 snssi 4725 . . . . . . . . . 10 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
76resabs1d 5871 . . . . . . . . 9 (𝐴𝐵 → ((𝐹𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
87eqcomd 2830 . . . . . . . 8 (𝐴𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹𝐵) ↾ {𝐴}))
98funeqd 6365 . . . . . . 7 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹𝐵) ↾ {𝐴})))
109biimpd 232 . . . . . 6 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹𝐵) ↾ {𝐴})))
115, 10anim12d 611 . . . . 5 (𝐴𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
1211impcom 411 . . . 4 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
13 df-dfat 43601 . . . . 5 ((𝐹𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
14 afvfundmfveq 43620 . . . . 5 ((𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)'''𝐴) = ((𝐹𝐵)‘𝐴))
1513, 14sylbir 238 . . . 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 6680 . . . 4 (𝐴𝐵 → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
1817adantl 485 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)‘𝐴) = (𝐹𝐴))
19 df-dfat 43601 . . . . . 6 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
20 afvfundmfveq 43620 . . . . . 6 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
2119, 20sylbir 238 . . . . 5 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = (𝐹𝐴))
2221eqcomd 2830 . . . 4 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹𝐴) = (𝐹'''𝐴))
2322adantr 484 . . 3 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → (𝐹𝐴) = (𝐹'''𝐴))
2416, 18, 233eqtrd 2863 . 2 (((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
25 pm3.4 809 . . . . . . . . . 10 ((𝐴𝐵𝐴 ∈ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
261, 25sylbi 220 . . . . . . . . 9 (𝐴 ∈ (𝐵 ∩ dom 𝐹) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2726, 3eleq2s 2934 . . . . . . . 8 (𝐴 ∈ dom (𝐹𝐵) → (𝐴𝐵𝐴 ∈ dom 𝐹))
2827com12 32 . . . . . . 7 (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) → 𝐴 ∈ dom 𝐹))
297funeqd 6365 . . . . . . . 8 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) ↔ Fun (𝐹 ↾ {𝐴})))
3029biimpd 232 . . . . . . 7 (𝐴𝐵 → (Fun ((𝐹𝐵) ↾ {𝐴}) → Fun (𝐹 ↾ {𝐴})))
3128, 30anim12d 611 . . . . . 6 (𝐴𝐵 → ((𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))))
3231con3d 155 . . . . 5 (𝐴𝐵 → (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
3332impcom 411 . . . 4 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ¬ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
34 afvnfundmuv 43621 . . . . 5 (¬ (𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)'''𝐴) = V)
3513, 34sylnbir 334 . . . 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 43621 . . . . . 6 𝐹 defAt 𝐴 → (𝐹'''𝐴) = V)
3819, 37sylnbir 334 . . . . 5 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹'''𝐴) = V)
3938eqcomd 2830 . . . 4 (¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → V = (𝐹'''𝐴))
4039adantr 484 . . 3 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → V = (𝐹'''𝐴))
4136, 40eqtrd 2859 . 2 ((¬ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) ∧ 𝐴𝐵) → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
4224, 41pm2.61ian 811 1 (𝐴𝐵 → ((𝐹𝐵)'''𝐴) = (𝐹'''𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cin 3918  {csn 4550  dom cdm 5542  cres 5544  Fun wfun 6337  cfv 6343   defAt wdfat 43598  '''cafv 43599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-aiota 43568  df-dfat 43601  df-afv 43602
This theorem is referenced by: (None)
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