Theorem afv2res 44618

Description: The value of a restricted function for an argument at which the function is defined. Analog to fvres 6775. (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 44498	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		elin 3899	. . . . . . . . . 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 5902	. . . . . . . . 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 412	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
7		snssi 4738	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
8	7	resabs1d 5911	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
9	8	eqcomd 2744	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
10	9	funeqd 6440	. . . . . . . 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 608	. . . . . 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 406	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
16		df-dfat 44498	. . . 4 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
17		dfatafv2iota 44589	. . . 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 3426	. . . . . 6 ⊢ 𝑥 ∈ V
21	20	brresi 5889	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴𝐹𝑥))
22	21	baib 535	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ 𝐴𝐹𝑥))
23	22	iotabidv 6402	. . 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 44589	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
26	25	eqcomd 2744	. . 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 2782	1 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ∩ cin 3882  {csn 4558   class class class wbr 5070  dom cdm 5580   ↾ cres 5582  ℩cio 6374  Fun wfun 6412   defAt wdfat 44495  ''''cafv2 44587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-res 5592  df-iota 6376  df-fun 6420  df-dfat 44498  df-afv2 44588

This theorem is referenced by: (None)