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 <-> A.x e. A E!y x Fy)

Distinct variable groups:   x,y,A   x, F,y

Proof of Theorem fnres

Step	Hyp	Ref	Expression
1		ancom 253	. . 3 |- ((A.x e. A E*y x Fy /\ A.x e. A E.y x Fy) <-> (A.x e. A E.y x Fy /\ A.x e. A E*y x Fy))
2		vex 2554	. . . . . . . . . 10 |- y e. _V
3	2	brres 4561	. . . . . . . . 9 |- (x( F |` A)y <-> (x Fy /\ x e. A))
4		ancom 253	. . . . . . . . 9 |- ((x Fy /\ x e. A) <-> (x e. A /\ x Fy))
5	3, 4	bitri 173	. . . . . . . 8 |- (x( F |` A)y <-> (x e. A /\ x Fy))
6	5	mobii 1934	. . . . . . 7 |- (E*y x( F |` A)y <-> E*y(x e. A /\ x Fy))
7		moanimv 1972	. . . . . . 7 |- (E*y(x e. A /\ x Fy) <-> (x e. A -> E*y x Fy))
8	6, 7	bitri 173	. . . . . 6 |- (E*y x( F |` A)y <-> (x e. A -> E*y x Fy))
9	8	albii 1356	. . . . 5 |- (A.xE*y x( F |` A)y <-> A.x(x e. A -> E*y x Fy))
10		relres 4582	. . . . . 6 |- Rel ( F |` A)
11		dffun6 4859	. . . . . 6 |- ( Fun ( F |` A) <-> ( Rel ( F |` A) /\ A.xE*y x( F |` A)y))
12	10, 11	mpbiran 846	. . . . 5 |- ( Fun ( F |` A) <-> A.xE*y x( F |` A)y)
13		df-ral 2305	. . . . 5 |- (A.x e. A E*y x Fy <-> A.x(x e. A -> E*y x Fy))
14	9, 12, 13	3bitr4i 201	. . . 4 |- ( Fun ( F |` A) <-> A.x e. A E*y x Fy)
15		dmres 4575	. . . . . . 7 |- dom ( F |` A) = (A i^i dom F)
16		inss1 3151	. . . . . . 7 |- (A i^i dom F) C_ A
17	15, 16	eqsstri 2969	. . . . . 6 |- dom ( F |` A) C_ A
18		eqss 2954	. . . . . 6 |- ( dom ( F |` A) = A <-> ( dom ( F |` A) C_ A /\ A C_ dom ( F |` A)))
19	17, 18	mpbiran 846	. . . . 5 |- ( dom ( F |` A) = A <-> A C_ dom ( F |` A))
20		dfss3 2929	. . . . . 6 |- (A C_ dom ( F |` A) <-> A.x e. A x e. dom ( F |` A))
21	15	elin2 3121	. . . . . . . . 9 |- (x e. dom ( F |` A) <-> (x e. A /\ x e. dom F))
22	21	baib 827	. . . . . . . 8 |- (x e. A -> (x e. dom ( F |` A) <-> x e. dom F))
23		vex 2554	. . . . . . . . 9 |- x e. _V
24	23	eldm 4475	. . . . . . . 8 |- (x e. dom F <-> E.y x Fy)
25	22, 24	syl6bb 185	. . . . . . 7 |- (x e. A -> (x e. dom ( F |` A) <-> E.y x Fy))
26	25	ralbiia 2332	. . . . . 6 |- (A.x e. A x e. dom ( F |` A) <-> A.x e. A E.y x Fy)
27	20, 26	bitri 173	. . . . 5 |- (A C_ dom ( F |` A) <-> A.x e. A E.y x Fy)
28	19, 27	bitri 173	. . . 4 |- ( dom ( F |` A) = A <-> A.x e. A E.y x Fy)
29	14, 28	anbi12i 433	. . 3 |- (( Fun ( F |` A) /\ dom ( F |` A) = A) <-> (A.x e. A E*y x Fy /\ A.x e. A E.y x Fy))
30		r19.26 2435	. . 3 |- (A.x e. A (E.y x Fy /\ E*y x Fy) <-> (A.x e. A E.y x Fy /\ A.x e. A E*y x Fy))
31	1, 29, 30	3bitr4i 201	. 2 |- (( Fun ( F |` A) /\ dom ( F |` A) = A) <-> A.x e. A (E.y x Fy /\ E*y x Fy))
32		df-fn 4848	. 2 |- (( F |` A) Fn A <-> ( Fun ( F |` A) /\ dom ( F |` A) = A))
33		eu5 1944	. . 3 |- (E!y x Fy <-> (E.y x Fy /\ E*y x Fy))
34	33	ralbii 2324	. 2 |- (A.x e. A E!y x Fy <-> A.x e. A (E.y x Fy /\ E*y x Fy))
35	31, 32, 34	3bitr4i 201	1 |- (( F |` A) Fn A <-> A.x e. A E!y x Fy)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390  E!weu 1897  E*wmo 1898  A.wral 2300   i^i cin 2910   C_ 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