Theorem fnres 4941

Description: An equivalence for functionality of a restriction. Compare dffun8 4855. (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 2538	. . . . . . . . . 10 ⊢ y ∈ V
3	2	brres 4545	. . . . . . . . 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 1919	. . . . . . 7 ⊢ (∃*y x(𝐹 ↾ A)y ↔ ∃*y(x ∈ A ∧ x𝐹y))
7		moanimv 1957	. . . . . . 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 1339	. . . . 5 ⊢ (∀x∃*y x(𝐹 ↾ A)y ↔ ∀x(x ∈ A → ∃*y x𝐹y))
10		relres 4566	. . . . . 6 ⊢ Rel (𝐹 ↾ A)
11		dffun6 4842	. . . . . 6 ⊢ (Fun (𝐹 ↾ A) ↔ (Rel (𝐹 ↾ A) ∧ ∀x∃*y x(𝐹 ↾ A)y))
12	10, 11	mpbiran 835	. . . . 5 ⊢ (Fun (𝐹 ↾ A) ↔ ∀x∃*y x(𝐹 ↾ A)y)
13		df-ral 2289	. . . . 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 4559	. . . . . . 7 ⊢ dom (𝐹 ↾ A) = (A ∩ dom 𝐹)
16		inss1 3134	. . . . . . 7 ⊢ (A ∩ dom 𝐹) ⊆ A
17	15, 16	eqsstri 2952	. . . . . 6 ⊢ dom (𝐹 ↾ A) ⊆ A
18		eqss 2937	. . . . . 6 ⊢ (dom (𝐹 ↾ A) = A ↔ (dom (𝐹 ↾ A) ⊆ A ∧ A ⊆ dom (𝐹 ↾ A)))
19	17, 18	mpbiran 835	. . . . 5 ⊢ (dom (𝐹 ↾ A) = A ↔ A ⊆ dom (𝐹 ↾ A))
20		dfss3 2912	. . . . . 6 ⊢ (A ⊆ dom (𝐹 ↾ A) ↔ ∀x ∈ A x ∈ dom (𝐹 ↾ A))
21	15	elin2 3104	. . . . . . . . 9 ⊢ (x ∈ dom (𝐹 ↾ A) ↔ (x ∈ A ∧ x ∈ dom 𝐹))
22	21	baib 818	. . . . . . . 8 ⊢ (x ∈ A → (x ∈ dom (𝐹 ↾ A) ↔ x ∈ dom 𝐹))
23		vex 2538	. . . . . . . . 9 ⊢ x ∈ V
24	23	eldm 4459	. . . . . . . 8 ⊢ (x ∈ dom 𝐹 ↔ ∃y x𝐹y)
25	22, 24	syl6bb 185	. . . . . . 7 ⊢ (x ∈ A → (x ∈ dom (𝐹 ↾ A) ↔ ∃y x𝐹y))
26	25	ralbiia 2316	. . . . . 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 436	. . 3 ⊢ ((Fun (𝐹 ↾ A) ∧ dom (𝐹 ↾ A) = A) ↔ (∀x ∈ A ∃*y x𝐹y ∧ ∀x ∈ A ∃y x𝐹y))
30		r19.26 2419	. . 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 4832	. 2 ⊢ ((𝐹 ↾ A) Fn A ↔ (Fun (𝐹 ↾ A) ∧ dom (𝐹 ↾ A) = A))
33		eu5 1929	. . 3 ⊢ (∃!y x𝐹y ↔ (∃y x𝐹y ∧ ∃*y x𝐹y))
34	33	ralbii 2308	. 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 1226   = wceq 1228  ∃wex 1362   ∈ wcel 1374  ∃!weu 1882  ∃*wmo 1883  ∀wral 2284   ∩ cin 2893   ⊆ wss 2894   class class class wbr 3738  dom cdm 4272   ↾ cres 4274  Rel wrel 4277  Fun wfun 4823   Fn wfn 4824

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-fun 4831  df-fn 4832

This theorem is referenced by:  f1ompt  5245