Theorem afv2res 47864

Description: The value of a restricted function for an argument at which the function is defined. Analog to fvres 6901. (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 47744	. . . . 5 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		elin 3929	. . . . . . . . . 10 ⊢ (𝐴 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹))
3	2	biimpri 231	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ (𝐵 ∩ dom 𝐹))
4		dmres 6012	. . . . . . . . 9 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
5	3, 4	eleqtrrdi 2880	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ dom 𝐹) → 𝐴 ∈ dom (𝐹 ↾ 𝐵))
6	5	ex 417	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ 𝐵)))
7		snssi 4756	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
8	7	resabs1d 6008	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐵 → ((𝐹 ↾ 𝐵) ↾ {𝐴}) = (𝐹 ↾ {𝐴}))
9	8	eqcomd 2775	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝐵 → (𝐹 ↾ {𝐴}) = ((𝐹 ↾ 𝐵) ↾ {𝐴}))
10	9	funeqd 6559	. . . . . . . 8 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) ↔ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
11	10	biimpd 232	. . . . . . 7 ⊢ (𝐴 ∈ 𝐵 → (Fun (𝐹 ↾ {𝐴}) → Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
12	6, 11	anim12d 620	. . . . . 6 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
13	12	com12 33	. . . . 5 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐴 ∈ 𝐵 → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
14	1, 13	sylbi 220	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐴 ∈ 𝐵 → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴}))))
15	14	imp 411	. . 3 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
16		df-dfat 47744	. . . 4 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 ↔ (𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})))
17		dfatafv2iota 47835	. . . 4 ⊢ ((𝐹 ↾ 𝐵) defAt 𝐴 → ((𝐹 ↾ 𝐵)''''𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥))
18	16, 17	sylbir 238	. . 3 ⊢ ((𝐴 ∈ dom (𝐹 ↾ 𝐵) ∧ Fun ((𝐹 ↾ 𝐵) ↾ {𝐴})) → ((𝐹 ↾ 𝐵)''''𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥))
19	15, 18	syl 18	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥))
20		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
21	20	brresi 5988	. . . . 5 ⊢ (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴𝐹𝑥))
22	21	baib 544	. . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴(𝐹 ↾ 𝐵)𝑥 ↔ 𝐴𝐹𝑥))
23	22	iotabidv 6521	. . 3 ⊢ (𝐴 ∈ 𝐵 → (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
24	23	adantl 486	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (℩𝑥𝐴(𝐹 ↾ 𝐵)𝑥) = (℩𝑥𝐴𝐹𝑥))
25		dfatafv2iota 47835	. . . 4 ⊢ (𝐹 defAt 𝐴 → (𝐹''''𝐴) = (℩𝑥𝐴𝐹𝑥))
26	25	eqcomd 2775	. . 3 ⊢ (𝐹 defAt 𝐴 → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
27	26	adantr 485	. 2 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → (℩𝑥𝐴𝐹𝑥) = (𝐹''''𝐴))
28	19, 24, 27	3eqtrd 2808	1 ⊢ ((𝐹 defAt 𝐴 ∧ 𝐴 ∈ 𝐵) → ((𝐹 ↾ 𝐵)''''𝐴) = (𝐹''''𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∩ cin 3912  {csn 4594   class class class wbr 5113  dom cdm 5662   ↾ cres 5664  ℩cio 6491  Fun wfun 6531   defAt wdfat 47741  ''''cafv2 47833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-dfat 47744  df-afv2 47834

This theorem is referenced by: (None)