Theorem fnres 5200

Description: An equivalence for functionality of a restriction. Compare dffun8 5135. (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 437	. . 3 ⊢ ((∀x ∈ A ∃*y xFy ∧ ∀x ∈ A ∃y xFy) ↔ (∀x ∈ A ∃y xFy ∧ ∀x ∈ A ∃*y xFy))
2		brres 4950	. . . . . . . . 9 ⊢ (x(F ↾ A)y ↔ (xFy ∧ x ∈ A))
3		ancom 437	. . . . . . . . 9 ⊢ ((xFy ∧ x ∈ A) ↔ (x ∈ A ∧ xFy))
4	2, 3	bitri 240	. . . . . . . 8 ⊢ (x(F ↾ A)y ↔ (x ∈ A ∧ xFy))
5	4	mobii 2240	. . . . . . 7 ⊢ (∃*y x(F ↾ A)y ↔ ∃*y(x ∈ A ∧ xFy))
6		moanimv 2262	. . . . . . 7 ⊢ (∃*y(x ∈ A ∧ xFy) ↔ (x ∈ A → ∃*y xFy))
7	5, 6	bitri 240	. . . . . 6 ⊢ (∃*y x(F ↾ A)y ↔ (x ∈ A → ∃*y xFy))
8	7	albii 1566	. . . . 5 ⊢ (∀x∃*y x(F ↾ A)y ↔ ∀x(x ∈ A → ∃*y xFy))
9		dffun6 5125	. . . . 5 ⊢ (Fun (F ↾ A) ↔ ∀x∃*y x(F ↾ A)y)
10		df-ral 2620	. . . . 5 ⊢ (∀x ∈ A ∃*y xFy ↔ ∀x(x ∈ A → ∃*y xFy))
11	8, 9, 10	3bitr4i 268	. . . 4 ⊢ (Fun (F ↾ A) ↔ ∀x ∈ A ∃*y xFy)
12		dmres 4987	. . . . . . 7 ⊢ dom (F ↾ A) = (A ∩ dom F)
13		inss1 3476	. . . . . . 7 ⊢ (A ∩ dom F) ⊆ A
14	12, 13	eqsstri 3302	. . . . . 6 ⊢ dom (F ↾ A) ⊆ A
15		eqss 3288	. . . . . 6 ⊢ (dom (F ↾ A) = A ↔ (dom (F ↾ A) ⊆ A ∧ A ⊆ dom (F ↾ A)))
16	14, 15	mpbiran 884	. . . . 5 ⊢ (dom (F ↾ A) = A ↔ A ⊆ dom (F ↾ A))
17		dfss3 3264	. . . . 5 ⊢ (A ⊆ dom (F ↾ A) ↔ ∀x ∈ A x ∈ dom (F ↾ A))
18	12	elin2 3447	. . . . . . . 8 ⊢ (x ∈ dom (F ↾ A) ↔ (x ∈ A ∧ x ∈ dom F))
19	18	baib 871	. . . . . . 7 ⊢ (x ∈ A → (x ∈ dom (F ↾ A) ↔ x ∈ dom F))
20		eldm 4899	. . . . . . 7 ⊢ (x ∈ dom F ↔ ∃y xFy)
21	19, 20	syl6bb 252	. . . . . 6 ⊢ (x ∈ A → (x ∈ dom (F ↾ A) ↔ ∃y xFy))
22	21	ralbiia 2647	. . . . 5 ⊢ (∀x ∈ A x ∈ dom (F ↾ A) ↔ ∀x ∈ A ∃y xFy)
23	16, 17, 22	3bitri 262	. . . 4 ⊢ (dom (F ↾ A) = A ↔ ∀x ∈ A ∃y xFy)
24	11, 23	anbi12i 678	. . 3 ⊢ ((Fun (F ↾ A) ∧ dom (F ↾ A) = A) ↔ (∀x ∈ A ∃*y xFy ∧ ∀x ∈ A ∃y xFy))
25		r19.26 2747	. . 3 ⊢ (∀x ∈ A (∃y xFy ∧ ∃*y xFy) ↔ (∀x ∈ A ∃y xFy ∧ ∀x ∈ A ∃*y xFy))
26	1, 24, 25	3bitr4i 268	. 2 ⊢ ((Fun (F ↾ A) ∧ dom (F ↾ A) = A) ↔ ∀x ∈ A (∃y xFy ∧ ∃*y xFy))
27		df-fn 4791	. 2 ⊢ ((F ↾ A) Fn A ↔ (Fun (F ↾ A) ∧ dom (F ↾ A) = A))
28		eu5 2242	. . 3 ⊢ (∃!y xFy ↔ (∃y xFy ∧ ∃*y xFy))
29	28	ralbii 2639	. 2 ⊢ (∀x ∈ A ∃!y xFy ↔ ∀x ∈ A (∃y xFy ∧ ∃*y xFy))
30	26, 27, 29	3bitr4i 268	1 ⊢ ((F ↾ A) Fn A ↔ ∀x ∈ A ∃!y xFy)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  ∀wral 2615   ∩ cin 3209   ⊆ wss 3258   class class class wbr 4640  dom cdm 4773   ↾ cres 4775  Fun wfun 4776   Fn wfn 4777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791

This theorem is referenced by: (None)