Theorem fnres 5199

Description: An equivalence for functionality of a restriction. Compare dffun8 5134. (Contributed by Mario Carneiro, 20-May-2015.)

Assertion

Ref	Expression
fnres	|- ((F |` A) Fn A <-> A.x e. A E!y xFy)

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

Proof of Theorem fnres

Step	Hyp	Ref	Expression
1		ancom 437	. . 3 |- ((A.x e. A E*y xFy /\ A.x e. A E.y xFy) <-> (A.x e. A E.y xFy /\ A.x e. A E*y xFy))
2		brres 4949	. . . . . . . . 9 |- (x(F |` A)y <-> (xFy /\ x e. A))
3		ancom 437	. . . . . . . . 9 |- ((xFy /\ x e. A) <-> (x e. A /\ xFy))
4	2, 3	bitri 240	. . . . . . . 8 |- (x(F |` A)y <-> (x e. A /\ xFy))
5	4	mobii 2240	. . . . . . 7 |- (E*y x(F |` A)y <-> E*y(x e. A /\ xFy))
6		moanimv 2262	. . . . . . 7 |- (E*y(x e. A /\ xFy) <-> (x e. A -> E*y xFy))
7	5, 6	bitri 240	. . . . . 6 |- (E*y x(F |` A)y <-> (x e. A -> E*y xFy))
8	7	albii 1566	. . . . 5 |- (A.xE*y x(F |` A)y <-> A.x(x e. A -> E*y xFy))
9		dffun6 5124	. . . . 5 |- (Fun (F |` A) <-> A.xE*y x(F |` A)y)
10		df-ral 2619	. . . . 5 |- (A.x e. A E*y xFy <-> A.x(x e. A -> E*y xFy))
11	8, 9, 10	3bitr4i 268	. . . 4 |- (Fun (F |` A) <-> A.x e. A E*y xFy)
12		dmres 4986	. . . . . . 7 |- dom (F |` A) = (A i^i dom F)
13		inss1 3475	. . . . . . 7 |- (A i^i dom F) C_ A
14	12, 13	eqsstri 3301	. . . . . 6 |- dom (F |` A) C_ A
15		eqss 3287	. . . . . 6 |- (dom (F |` A) = A <-> (dom (F |` A) C_ A /\ A C_ dom (F |` A)))
16	14, 15	mpbiran 884	. . . . 5 |- (dom (F |` A) = A <-> A C_ dom (F |` A))
17		dfss3 3263	. . . . 5 |- (A C_ dom (F |` A) <-> A.x e. A x e. dom (F |` A))
18	12	elin2 3446	. . . . . . . 8 |- (x e. dom (F |` A) <-> (x e. A /\ x e. dom F))
19	18	baib 871	. . . . . . 7 |- (x e. A -> (x e. dom (F |` A) <-> x e. dom F))
20		eldm 4898	. . . . . . 7 |- (x e. dom F <-> E.y xFy)
21	19, 20	syl6bb 252	. . . . . 6 |- (x e. A -> (x e. dom (F |` A) <-> E.y xFy))
22	21	ralbiia 2646	. . . . 5 |- (A.x e. A x e. dom (F |` A) <-> A.x e. A E.y xFy)
23	16, 17, 22	3bitri 262	. . . 4 |- (dom (F |` A) = A <-> A.x e. A E.y xFy)
24	11, 23	anbi12i 678	. . 3 |- ((Fun (F |` A) /\ dom (F |` A) = A) <-> (A.x e. A E*y xFy /\ A.x e. A E.y xFy))
25		r19.26 2746	. . 3 |- (A.x e. A (E.y xFy /\ E*y xFy) <-> (A.x e. A E.y xFy /\ A.x e. A E*y xFy))
26	1, 24, 25	3bitr4i 268	. 2 |- ((Fun (F |` A) /\ dom (F |` A) = A) <-> A.x e. A (E.y xFy /\ E*y xFy))
27		df-fn 4790	. 2 |- ((F |` A) Fn A <-> (Fun (F |` A) /\ dom (F |` A) = A))
28		eu5 2242	. . 3 |- (E!y xFy <-> (E.y xFy /\ E*y xFy))
29	28	ralbii 2638	. 2 |- (A.x e. A E!y xFy <-> A.x e. A (E.y xFy /\ E*y xFy))
30	26, 27, 29	3bitr4i 268	1 |- ((F |` A) Fn A <-> A.x e. A E!y xFy)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  E!weu 2204  E*wmo 2205  A.wral 2614   i^i cin 3208   C_ wss 3257   class class class wbr 4639  dom cdm 4772   |` cres 4774  Fun wfun 4775   Fn wfn 4776

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790

This theorem is referenced by: (None)