Theorem dfdfat2 38627

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 38611	. 2 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
2		relres 5131	. . . 4 |- Rel ( F |` { A } )
3		dffun8 5608	. . . 4 |- ( Fun ( F |` { A } ) <-> ( Rel ( F |` { A } ) /\ A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y ) )
4	2, 3	mpbiran 928	. . 3 |- ( Fun ( F |` { A } ) <-> A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y )
5	4	anbi2i 699	. 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 3047	. . . . . . . 8 |- y e. _V
7	6	brres 5110	. . . . . . 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 2318	. . . . 5 |- ( A e. dom F -> ( E! y x ( F |` { A } ) y <-> E! y ( x F y /\ x e. { A } ) ) )
10	9	ralbidv 2826	. . . 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 5143	. . . . 5 |- ( A e. dom F -> A e. dom ( F |` { A } ) )
12		eldmressn 38618	. . . . 5 |- ( x e. dom ( F |` { A } ) -> x = A )
13		breq1 4404	. . . . . . . 8 |- ( x = A -> ( x F y <-> A F y ) )
14	13	anbi1d 710	. . . . . . 7 |- ( x = A -> ( ( x F y /\ x e. { A } ) <-> ( A F y /\ x e. { A } ) ) )
15		elsn 3981	. . . . . . . . 9 |- ( x e. { A } <-> x = A )
16	15	biimpri 210	. . . . . . . 8 |- ( x = A -> x e. { A } )
17	16	biantrud 510	. . . . . . 7 |- ( x = A -> ( A F y <-> ( A F y /\ x e. { A } ) ) )
18	14, 17	bitr4d 260	. . . . . 6 |- ( x = A -> ( ( x F y /\ x e. { A } ) <-> A F y ) )
19	18	eubidv 2318	. . . . 5 |- ( x = A -> ( E! y ( x F y /\ x e. { A } ) <-> E! y A F y ) )
20	11, 12, 19	ralbinrald 38614	. . . 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 257	. . 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 642	. 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 275	1 |- ( F defAt A <-> ( A e. dom F /\ E! y A F y ) )

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   E!weu 2298   A.wral 2736   {csn 3967   class class class wbr 4401   dom cdm 4833   |` cres 4835   Rel wrel 4838   Fun wfun 5575   defAt wdfat 38608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-res 4845  df-fun 5583  df-dfat 38611

This theorem is referenced by:  afveu  38649  rlimdmafv  38673