Theorem fnres 6613

Description: An equivalence for functionality of a restriction. Compare dffun8 6514. (Contributed by Mario Carneiro, 20-May-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)

Assertion

Ref	Expression
fnres	⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Proof of Theorem fnres

Step	Hyp	Ref	Expression
1		ancom 460	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦))
2		vex 3442	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	2	brresi 5943	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
4	3	mobii 2541	. . . . . . 7 ⊢ (∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
5		moanimv 2612	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
6	4, 5	bitri 275	. . . . . 6 ⊢ (∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
7	6	albii 1819	. . . . 5 ⊢ (∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
8		relres 5960	. . . . . 6 ⊢ Rel (𝐹 ↾ 𝐴)
9		dffun6 6497	. . . . . 6 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ (Rel (𝐹 ↾ 𝐴) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦))
10	8, 9	mpbiran 709	. . . . 5 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦)
11		df-ral 3045	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
12	7, 10, 11	3bitr4i 303	. . . 4 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦)
13		dmres 5967	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
14		inss1 4190	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
15	13, 14	eqsstri 3984	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
16		eqss 3953	. . . . . 6 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ dom (𝐹 ↾ 𝐴)))
17	15, 16	mpbiran 709	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ 𝐴 ⊆ dom (𝐹 ↾ 𝐴))
18		dfss3 3926	. . . . . 6 ⊢ (𝐴 ⊆ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom (𝐹 ↾ 𝐴))
19	13	elin2 4156	. . . . . . . . 9 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐹))
20	19	baib 535	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ 𝑥 ∈ dom 𝐹))
21		vex 3442	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22	21	eldm 5847	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
23	20, 22	bitrdi 287	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ ∃𝑦 𝑥𝐹𝑦))
24	23	ralbiia 3073	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
25	18, 24	bitri 275	. . . . 5 ⊢ (𝐴 ⊆ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
26	17, 25	bitri 275	. . . 4 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
27	12, 26	anbi12i 628	. . 3 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦))
28		r19.26 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦))
29	1, 27, 28	3bitr4i 303	. 2 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
30		df-fn 6489	. 2 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴))
31		df-eu 2562	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
32	31	ralbii 3075	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
33	29, 30, 32	3bitr4i 303	1 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∃!weu 2561  ∀wral 3044   ∩ cin 3904   ⊆ wss 3905   class class class wbr 5095  dom cdm 5623   ↾ cres 5625  Rel wrel 5628  Fun wfun 6480   Fn wfn 6481

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 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-res 5635  df-fun 6488  df-fn 6489

This theorem is referenced by:  f1ompt  7049  omxpenlem  9002  tz6.12-afv  47161  tz6.12-afv2  47228