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Theorem afv2res 45545
Description: The value of a restricted function for an argument at which the function is defined. Analog to fvres 6866. (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 45425 . . . . 5 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2 elin 3931 . . . . . . . . . 10 (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴𝐵𝐴 ∈ dom 𝐹))
32biimpri 227 . . . . . . . . 9 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
4 dmres 5964 . . . . . . . . 9 dom (𝐹𝐵) = (𝐵 ∩ dom 𝐹)
53, 4eleqtrrdi 2849 . . . . . . . 8 ((𝐴𝐵𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹𝐵))
65ex 414 . . . . . . 7 (𝐴𝐵 → (𝐴 ∈ dom 𝐹𝐴 ∈ dom (𝐹𝐵)))
7 snssi 4773 . . . . . . . . . . 11 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
87resabs1d 5973 . . . . . . . . . 10 (𝐴𝐵 → ((𝐹𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
98eqcomd 2743 . . . . . . . . 9 (𝐴𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹𝐵) ↾ {𝐴}))
109funeqd 6528 . . . . . . . 8 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹𝐵) ↾ {𝐴})))
1110biimpd 228 . . . . . . 7 (𝐴𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹𝐵) ↾ {𝐴})))
126, 11anim12d 610 . . . . . 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 216 . . . 4 (𝐹 defAt 𝐴 → (𝐴𝐵 → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴}))))
1514imp 408 . . 3 ((𝐹 defAt 𝐴𝐴𝐵) → (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
16 df-dfat 45425 . . . 4 ((𝐹𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹𝐵) ∧ Fun ((𝐹𝐵) ↾ {𝐴})))
17 dfatafv2iota 45516 . . . 4 ((𝐹𝐵) defAt 𝐴 → ((𝐹𝐵)''''𝐴) = (℩𝑥𝐴(𝐹𝐵)𝑥))
1816, 17sylbir 234 . . 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 3452 . . . . . 6 𝑥 ∈ V
2120brresi 5951 . . . . 5 (𝐴(𝐹𝐵)𝑥 ↔ (𝐴𝐵𝐴𝐹𝑥))
2221baib 537 . . . 4 (𝐴𝐵 → (𝐴(𝐹𝐵)𝑥𝐴𝐹𝑥))
2322iotabidv 6485 . . 3 (𝐴𝐵 → (℩𝑥𝐴(𝐹𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
2423adantl 483 . 2 ((𝐹 defAt 𝐴𝐴𝐵) → (℩𝑥𝐴(𝐹𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
25 dfatafv2iota 45516 . . . 4 (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
2625eqcomd 2743 . . 3 (𝐹 defAt 𝐴 → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
2726adantr 482 . 2 ((𝐹 defAt 𝐴𝐴𝐵) → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
2819, 24, 273eqtrd 2781 1 ((𝐹 defAt 𝐴𝐴𝐵) → ((𝐹𝐵)''''𝐴) = (𝐹''''𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cin 3914  {csn 4591   class class class wbr 5110  dom cdm 5638  cres 5640  cio 6451  Fun wfun 6495   defAt wdfat 45422  ''''cafv2 45514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-dfat 45425  df-afv2 45515
This theorem is referenced by: (None)
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