Theorem afv2res 44731

Description: The value of a restricted function for an argument at which the function is defined. Analog to fvres 6793. (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 44611	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		elin 3903	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹))
3	2	biimpri 227	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
4		dmres 5913	. . . . . . . . 9 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5	3, 4	eleqtrrdi 2850	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹 ↾ 𝐵))
6	5	ex 413	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
7		snssi 4741	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
8	7	resabs1d 5922	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
9	8	eqcomd 2744	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
10	9	funeqd 6456	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
11	10	biimpd 228	. . . . . . 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 216	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐴 ∈ 𝐵 → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
15	14	imp 407	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
16		df-dfat 44611	. . . 4 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
17		dfatafv2iota 44702	. . . 4 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)''''𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥))
18	16, 17	sylbir 234	. . 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 3436	. . . . . 6 ⊢ 𝑥 ∈ V
21	20	brresi 5900	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴𝐹𝑥))
22	21	baib 536	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ 𝐴𝐹𝑥))
23	22	iotabidv 6417	. . 3 ⊢ (𝐴 ∈ 𝐵 → (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
24	23	adantl 482	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
25		dfatafv2iota 44702	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
26	25	eqcomd 2744	. . 3 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
27	26	adantr 481	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
28	19, 24, 27	3eqtrd 2782	1 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∩ cin 3886  {csn 4561   class class class wbr 5074  dom cdm 5589   ↾ cres 5591  ℩cio 6389  Fun wfun 6427   defAt wdfat 44608  ''''cafv2 44700

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6391  df-fun 6435  df-dfat 44611  df-afv2 44701

This theorem is referenced by: (None)