Theorem fnres 5474

Description: An equivalence for functionality of a restriction. Compare dffun8 5379. (Contributed by Mario Carneiro, 20-May-2015.)

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 266	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦))
2		vex 2815	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
3	2	brres 5043	. . . . . . . . 9 ⊢ (𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥𝐹𝑦 ∧ 𝑥 ∈ 𝐴))
4		ancom 266	. . . . . . . . 9 ⊢ ((𝑥𝐹𝑦 ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
5	3, 4	bitri 184	. . . . . . . 8 ⊢ (𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
6	5	mobii 2117	. . . . . . 7 ⊢ (∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦))
7		moanimv 2156	. . . . . . 7 ⊢ (∃*𝑦(𝑥 ∈ 𝐴 ∧ 𝑥𝐹𝑦) ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
8	6, 7	bitri 184	. . . . . 6 ⊢ (∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
9	8	albii 1519	. . . . 5 ⊢ (∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
10		relres 5065	. . . . . 6 ⊢ Rel (𝐹 ↾ 𝐴)
11		dffun6 5365	. . . . . 6 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ (Rel (𝐹 ↾ 𝐴) ∧ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦))
12	10, 11	mpbiran 949	. . . . 5 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ ∀𝑥∃*𝑦 𝑥(𝐹 ↾ 𝐴)𝑦)
13		df-ral 2525	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∃*𝑦 𝑥𝐹𝑦))
14	9, 12, 13	3bitr4i 212	. . . 4 ⊢ (Fun (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦)
15		dmres 5058	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹)
16		inss1 3440	. . . . . . 7 ⊢ (𝐴 ∩ dom 𝐹) ⊆ 𝐴
17	15, 16	eqsstri 3269	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐴) ⊆ 𝐴
18		eqss 3252	. . . . . 6 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ (dom (𝐹 ↾ 𝐴) ⊆ 𝐴 ∧ 𝐴 ⊆ dom (𝐹 ↾ 𝐴)))
19	17, 18	mpbiran 949	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ 𝐴 ⊆ dom (𝐹 ↾ 𝐴))
20		dfss3 3226	. . . . . 6 ⊢ (𝐴 ⊆ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ dom (𝐹 ↾ 𝐴))
21	15	elin2 3406	. . . . . . . . 9 ⊢ (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ dom 𝐹))
22	21	baib 927	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ 𝑥 ∈ dom 𝐹))
23		vex 2815	. . . . . . . . 9 ⊢ 𝑥 ∈ V
24	23	eldm 4952	. . . . . . . 8 ⊢ (𝑥 ∈ dom 𝐹 ↔ ∃𝑦 𝑥𝐹𝑦)
25	22, 24	bitrdi 196	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ ∃𝑦 𝑥𝐹𝑦))
26	25	ralbiia 2556	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
27	20, 26	bitri 184	. . . . 5 ⊢ (𝐴 ⊆ dom (𝐹 ↾ 𝐴) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
28	19, 27	bitri 184	. . . 4 ⊢ (dom (𝐹 ↾ 𝐴) = 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦)
29	14, 28	anbi12i 460	. . 3 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴) ↔ (∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦))
30		r19.26 2669	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦 𝑥𝐹𝑦 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 𝑥𝐹𝑦))
31	1, 29, 30	3bitr4i 212	. 2 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴) ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
32		df-fn 5354	. 2 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ (Fun (𝐹 ↾ 𝐴) ∧ dom (𝐹 ↾ 𝐴) = 𝐴))
33		eu5 2128	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
34	33	ralbii 2548	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦 ↔ ∀𝑥 ∈ 𝐴 (∃𝑦 𝑥𝐹𝑦 ∧ ∃*𝑦 𝑥𝐹𝑦))
35	31, 32, 34	3bitr4i 212	1 ⊢ ((𝐹 ↾ 𝐴) Fn 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑥𝐹𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541  ∃!weu 2080  ∃*wmo 2081   ∈ wcel 2203  ∀wral 2520   ∩ cin 3209   ⊆ wss 3210   class class class wbr 4108  dom cdm 4748   ↾ cres 4750  Rel wrel 4753  Fun wfun 5345   Fn wfn 5346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-res 4760  df-fun 5353  df-fn 5354

This theorem is referenced by:  f1ompt  5827