Theorem afv2res 47244

Description: The value of a restricted function for an argument at which the function is defined. Analog to fvres 6880. (Contributed by AV, 5-Sep-2022.)

Assertion

Ref	Expression
afv2res	⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))

Proof of Theorem afv2res

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dfat 47124	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		elin 3933	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹))
3	2	biimpri 228	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
4		dmres 5986	. . . . . . . . 9 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5	3, 4	eleqtrrdi 2840	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹 ↾ 𝐵))
6	5	ex 412	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
7		snssi 4775	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
8	7	resabs1d 5982	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
9	8	eqcomd 2736	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
10	9	funeqd 6541	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
11	10	biimpd 229	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
12	6, 11	anim12d 609	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
13	12	com12 32	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ 𝐵 → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
14	1, 13	sylbi 217	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐴 ∈ 𝐵 → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
15	14	imp 406	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
16		df-dfat 47124	. . . 4 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
17		dfatafv2iota 47215	. . . 4 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)''''𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥))
18	16, 17	sylbir 235	. . 3 ⊢ ((𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → ((𝐹 ↾ 𝐵)''''𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥))
19	15, 18	syl 17	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥))
20		vex 3454	. . . . . 6 ⊢ 𝑥 ∈ V
21	20	brresi 5962	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴𝐹𝑥))
22	21	baib 535	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ 𝐴𝐹𝑥))
23	22	iotabidv 6498	. . 3 ⊢ (𝐴 ∈ 𝐵 → (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
24	23	adantl 481	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
25		dfatafv2iota 47215	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
26	25	eqcomd 2736	. . 3 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
27	26	adantr 480	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
28	19, 24, 27	3eqtrd 2769	1 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3916  {csn 4592   class class class wbr 5110  dom cdm 5641   ↾ cres 5643  ℩cio 6465  Fun wfun 6508   defAt wdfat 47121  ''''cafv2 47213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-res 5653  df-iota 6467  df-fun 6516  df-dfat 47124  df-afv2 47214

This theorem is referenced by: (None)