Theorem fnres 6619

Description: An equivalence for functionality of a restriction. Compare dffun8 6520. (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 3434	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3	2	brresi 5947	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
4	3	mobii 2549	. . . . . . 7 ⊢ (∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
5		moanimv 2620	. . . . . . 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 1821	. . . . 5 ⊢ (∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
8		relres 5964	. . . . . 6 ⊢ Rel (𝐹 ↾ 𝐴)
9		dffun6 6503	. . . . . 6 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ (Rel (𝐹 ↾ 𝐴) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦))
10	8, 9	mpbiran 710	. . . . 5 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦)
11		df-ral 3053	. . . . 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 5971	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
14		inss1 4178	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
15	13, 14	eqsstri 3969	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
16		eqss 3938	. . . . . 6 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ dom (𝐹 ↾ 𝐴)))
17	15, 16	mpbiran 710	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ 𝐴 ⊆ dom (𝐹 ↾ 𝐴))
18		dfss3 3911	. . . . . 6 ⊢ (𝐴 ⊆ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom (𝐹 ↾ 𝐴))
19	13	elin2 4144	. . . . . . . . 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 3434	. . . . . . . . 9 ⊢ 𝑥 ∈ V
22	21	eldm 5849	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
23	20, 22	bitrdi 287	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ ∃𝑦 𝑥𝐹𝑦))
24	23	ralbiia 3082	. . . . . 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 629	. . 3 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦))
28		r19.26 3098	. . 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 6495	. 2 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴))
31		df-eu 2570	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
32	31	ralbii 3084	. 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 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2538  ∃!weu 2569  ∀wral 3052   ∩ cin 3889   ⊆ wss 3890   class class class wbr 5086  dom cdm 5624   ↾ cres 5626  Rel wrel 5629  Fun wfun 6486   Fn wfn 6487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-fun 6494  df-fn 6495

This theorem is referenced by:  f1ompt  7057  omxpenlem  9009  noinfepregs  35293  tz6.12-afv  47633  tz6.12-afv2  47700