Theorem dfdfat2 30037

Description: Alternate definition of the predicate "defined at" not using the Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)

Assertion

Ref	Expression
dfdfat2	|- ( F defAt A <-> ( A e. dom F /\ E! y A F y ) )

Distinct variable groups:   y, A   y, F

Proof of Theorem dfdfat2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-dfat 30020	. 2 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
2		relres 5138	. . . 4 |- Rel ( F |` { A } )
3		dffun8 5445	. . . 4 |- ( Fun ( F |` { A } ) <-> ( Rel ( F |` { A } ) /\ A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y ) )
4	2, 3	mpbiran 909	. . 3 |- ( Fun ( F |` { A } ) <-> A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y )
5	4	anbi2i 694	. 2 |- ( ( A e. dom F /\ Fun ( F |` { A } ) ) <-> ( A e. dom F /\ A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y ) )
6		vex 2975	. . . . . . . 8 |- y e. _V
7	6	brres 5117	. . . . . . 7 |- ( x ( F |` { A } ) y <-> ( x F y /\ x e. { A } ) )
8	7	a1i 11	. . . . . 6 |- ( A e. dom F -> ( x ( F |` { A } ) y <-> ( x F y /\ x e. { A } ) ) )
9	8	eubidv 2276	. . . . 5 |- ( A e. dom F -> ( E! y x ( F |` { A } ) y <-> E! y ( x F y /\ x e. { A } ) ) )
10	9	ralbidv 2735	. . . 4 |- ( A e. dom F -> ( A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y <-> A. x e. dom ( F |` { A } ) E! y ( x F y /\ x e. { A } ) ) )
11		eldmressnsn 30029	. . . . 5 |- ( A e. dom F -> A e. dom ( F |` { A } ) )
12		eldmressn 30027	. . . . 5 |- ( x e. dom ( F |` { A } ) -> x = A )
13		breq1 4295	. . . . . . . 8 |- ( x = A -> ( x F y <-> A F y ) )
14	13	anbi1d 704	. . . . . . 7 |- ( x = A -> ( ( x F y /\ x e. { A } ) <-> ( A F y /\ x e. { A } ) ) )
15		elsn 3891	. . . . . . . . 9 |- ( x e. { A } <-> x = A )
16	15	biimpri 206	. . . . . . . 8 |- ( x = A -> x e. { A } )
17	16	biantrud 507	. . . . . . 7 |- ( x = A -> ( A F y <-> ( A F y /\ x e. { A } ) ) )
18	14, 17	bitr4d 256	. . . . . 6 |- ( x = A -> ( ( x F y /\ x e. { A } ) <-> A F y ) )
19	18	eubidv 2276	. . . . 5 |- ( x = A -> ( E! y ( x F y /\ x e. { A } ) <-> E! y A F y ) )
20	11, 12, 19	ralbinrald 30023	. . . 4 |- ( A e. dom F -> ( A. x e. dom ( F |` { A } ) E! y ( x F y /\ x e. { A } ) <-> E! y A F y ) )
21	10, 20	bitrd 253	. . 3 |- ( A e. dom F -> ( A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y <-> E! y A F y ) )
22	21	pm5.32i 637	. 2 |- ( ( A e. dom F /\ A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y ) <-> ( A e. dom F /\ E! y A F y ) )
23	1, 5, 22	3bitri 271	1 |- ( F defAt A <-> ( A e. dom F /\ E! y A F y ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   E!weu 2253   A.wral 2715   {csn 3877   class class class wbr 4292   dom cdm 4840   |` cres 4842   Rel wrel 4845   Fun wfun 5412   defAt wdfat 30017

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-res 4852  df-fun 5420  df-dfat 30020

This theorem is referenced by:  afveu  30059  rlimdmafv  30083