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Theorem afv2res 43721
Description: The value of a restricted function for an argument at which the function is defined. Analog to fvres 6680. (Contributed by AV, 5-Sep-2022.)
Assertion
Ref Expression
afv2res ((𝐹 defAt 𝐴𝐴𝐵) → ((𝐹𝐵)''''𝐴) = (𝐹''''𝐴))

Proof of Theorem afv2res
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-dfat 43601 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 elin 3935 . . . . . . . . . 10 (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴𝐵𝐴 ∈ dom 𝐹))
32biimpri 231 . . . . . . . . 9 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
4 dmres 5862 . . . . . . . . 9 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
53, 4eleqtrrdi 2927 . . . . . . . 8 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹𝐵))
65ex 416 . . . . . . 7 (𝐴𝐵 → (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹𝐵)))
7 snssi 4725 . . . . . . . . . . 11 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
87resabs1d 5871 . . . . . . . . . 10 (𝐴𝐵 → ((𝐹𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
98eqcomd 2830 . . . . . . . . 9 (𝐴𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹𝐵) ↾ {𝐴}))
109funeqd 6365 . . . . . . . 8 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹𝐵) ↾ {𝐴})))
1110biimpd 232 . . . . . . 7 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹𝐵) ↾ {𝐴})))
126, 11anim12d 611 . . . . . 6 (𝐴𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
1312com12 32 . . . . 5 ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
141, 13sylbi 220 . . . 4 (𝐹 defAt 𝐴 → (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
1514imp 410 . . 3 ((𝐹 defAt 𝐴𝐴𝐵) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
16 df-dfat 43601 . . . 4 ((𝐹𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
17 dfatafv2iota 43692 . . . 4 ((𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)''''𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥))
1816, 17sylbir 238 . . 3 ((𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})) → ((𝐹𝐵)''''𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥))
1915, 18syl 17 . 2 ((𝐹 defAt 𝐴𝐴𝐵) → ((𝐹𝐵)''''𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥))
20 vex 3483 . . . . . 6 𝑥 ∈ V
2120brresi 5849 . . . . 5 (𝐴(𝐹𝐵)𝑥 ↔ (𝐴𝐵𝐴𝐹𝑥))
2221baib 539 . . . 4 (𝐴𝐵 → (𝐴(𝐹𝐵)𝑥𝐴𝐹𝑥))
2322iotabidv 6327 . . 3 (𝐴𝐵 → (℩𝑥𝐴(𝐹𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
2423adantl 485 . 2 ((𝐹 defAt 𝐴𝐴𝐵) → (℩𝑥𝐴(𝐹𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
25 dfatafv2iota 43692 . . . 4 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2625eqcomd 2830 . . 3 (𝐹 defAt 𝐴 → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
2726adantr 484 . 2 ((𝐹 defAt 𝐴𝐴𝐵) → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
2819, 24, 273eqtrd 2863 1 ((𝐹 defAt 𝐴𝐴𝐵) → ((𝐹𝐵)''''𝐴) = (𝐹''''𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  cin 3918  {csn 4550   class class class wbr 5052  dom cdm 5542  cres 5544  cio 6300  Fun wfun 6337   defAt wdfat 43598  ''''cafv2 43690
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-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  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-br 5053  df-opab 5115  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-dfat 43601  df-afv2 43691
This theorem is referenced by: (None)
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