Theorem fnres 4958

Description: An equivalence for functionality of a restriction. Compare dffun8 4872. (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 253	. . 3 ⊢ ((∀x ∈ A ∃*y x𝐹y ∧ ∀x ∈ A ∃y x𝐹y) ↔ (∀x ∈ A ∃y x𝐹y ∧ ∀x ∈ A ∃*y x𝐹y))
2		vex 2554	. . . . . . . . . 10 ⊢ y ∈ V
3	2	brres 4561	. . . . . . . . 9 ⊢ (x(𝐹 ↾ A)y ↔ (x𝐹y ∧ x ∈ A))
4		ancom 253	. . . . . . . . 9 ⊢ ((x𝐹y ∧ x ∈ A) ↔ (x ∈ A ∧ x𝐹y))
5	3, 4	bitri 173	. . . . . . . 8 ⊢ (x(𝐹 ↾ A)y ↔ (x ∈ A ∧ x𝐹y))
6	5	mobii 1934	. . . . . . 7 ⊢ (∃*y x(𝐹 ↾ A)y ↔ ∃*y(x ∈ A ∧ x𝐹y))
7		moanimv 1972	. . . . . . 7 ⊢ (∃*y(x ∈ A ∧ x𝐹y) ↔ (x ∈ A → ∃*y x𝐹y))
8	6, 7	bitri 173	. . . . . 6 ⊢ (∃*y x(𝐹 ↾ A)y ↔ (x ∈ A → ∃*y x𝐹y))
9	8	albii 1356	. . . . 5 ⊢ (∀x∃*y x(𝐹 ↾ A)y ↔ ∀x(x ∈ A → ∃*y x𝐹y))
10		relres 4582	. . . . . 6 ⊢ Rel (𝐹 ↾ A)
11		dffun6 4859	. . . . . 6 ⊢ (Fun (𝐹 ↾ A) ↔ (Rel (𝐹 ↾ A) ∧ ∀x∃*y x(𝐹 ↾ A)y))
12	10, 11	mpbiran 846	. . . . 5 ⊢ (Fun (𝐹 ↾ A) ↔ ∀x∃*y x(𝐹 ↾ A)y)
13		df-ral 2305	. . . . 5 ⊢ (∀x ∈ A ∃*y x𝐹y ↔ ∀x(x ∈ A → ∃*y x𝐹y))
14	9, 12, 13	3bitr4i 201	. . . 4 ⊢ (Fun (𝐹 ↾ A) ↔ ∀x ∈ A ∃*y x𝐹y)
15		dmres 4575	. . . . . . 7 ⊢ dom (𝐹 ↾ A) = (A ∩ dom 𝐹)
16		inss1 3151	. . . . . . 7 ⊢ (A ∩ dom 𝐹) ⊆ A
17	15, 16	eqsstri 2969	. . . . . 6 ⊢ dom (𝐹 ↾ A) ⊆ A
18		eqss 2954	. . . . . 6 ⊢ (dom (𝐹 ↾ A) = A ↔ (dom (𝐹 ↾ A) ⊆ A ∧ A ⊆ dom (𝐹 ↾ A)))
19	17, 18	mpbiran 846	. . . . 5 ⊢ (dom (𝐹 ↾ A) = A ↔ A ⊆ dom (𝐹 ↾ A))
20		dfss3 2929	. . . . . 6 ⊢ (A ⊆ dom (𝐹 ↾ A) ↔ ∀x ∈ A x ∈ dom (𝐹 ↾ A))
21	15	elin2 3121	. . . . . . . . 9 ⊢ (x ∈ dom (𝐹 ↾ A) ↔ (x ∈ A ∧ x ∈ dom 𝐹))
22	21	baib 827	. . . . . . . 8 ⊢ (x ∈ A → (x ∈ dom (𝐹 ↾ A) ↔ x ∈ dom 𝐹))
23		vex 2554	. . . . . . . . 9 ⊢ x ∈ V
24	23	eldm 4475	. . . . . . . 8 ⊢ (x ∈ dom 𝐹 ↔ ∃y x𝐹y)
25	22, 24	syl6bb 185	. . . . . . 7 ⊢ (x ∈ A → (x ∈ dom (𝐹 ↾ A) ↔ ∃y x𝐹y))
26	25	ralbiia 2332	. . . . . 6 ⊢ (∀x ∈ A x ∈ dom (𝐹 ↾ A) ↔ ∀x ∈ A ∃y x𝐹y)
27	20, 26	bitri 173	. . . . 5 ⊢ (A ⊆ dom (𝐹 ↾ A) ↔ ∀x ∈ A ∃y x𝐹y)
28	19, 27	bitri 173	. . . 4 ⊢ (dom (𝐹 ↾ A) = A ↔ ∀x ∈ A ∃y x𝐹y)
29	14, 28	anbi12i 433	. . 3 ⊢ ((Fun (𝐹 ↾ A) ∧ dom (𝐹 ↾ A) = A) ↔ (∀x ∈ A ∃*y x𝐹y ∧ ∀x ∈ A ∃y x𝐹y))
30		r19.26 2435	. . 3 ⊢ (∀x ∈ A (∃y x𝐹y ∧ ∃*y x𝐹y) ↔ (∀x ∈ A ∃y x𝐹y ∧ ∀x ∈ A ∃*y x𝐹y))
31	1, 29, 30	3bitr4i 201	. 2 ⊢ ((Fun (𝐹 ↾ A) ∧ dom (𝐹 ↾ A) = A) ↔ ∀x ∈ A (∃y x𝐹y ∧ ∃*y x𝐹y))
32		df-fn 4848	. 2 ⊢ ((𝐹 ↾ A) Fn A ↔ (Fun (𝐹 ↾ A) ∧ dom (𝐹 ↾ A) = A))
33		eu5 1944	. . 3 ⊢ (∃!y x𝐹y ↔ (∃y x𝐹y ∧ ∃*y x𝐹y))
34	33	ralbii 2324	. 2 ⊢ (∀x ∈ A ∃!y x𝐹y ↔ ∀x ∈ A (∃y x𝐹y ∧ ∃*y x𝐹y))
35	31, 32, 34	3bitr4i 201	1 ⊢ ((𝐹 ↾ A) Fn A ↔ ∀x ∈ A ∃!y x𝐹y)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃!weu 1897  ∃*wmo 1898  ∀wral 2300   ∩ cin 2910   ⊆ wss 2911   class class class wbr 3755  dom cdm 4288   ↾ cres 4290  Rel wrel 4293  Fun wfun 4839   Fn wfn 4840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-fun 4847  df-fn 4848

This theorem is referenced by:  f1ompt  5263