Theorem dfdfat2 41211

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 41196	. 2 |- ( F defAt A <-> ( A e. dom F /\ Fun ( F |` { A } ) ) )
2		relres 5426	. . . 4 |- Rel ( F |` { A } )
3		dffun8 5916	. . . 4 |- ( Fun ( F |` { A } ) <-> ( Rel ( F |` { A } ) /\ A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y ) )
4	2, 3	mpbiran 953	. . 3 |- ( Fun ( F |` { A } ) <-> A. x e. dom ( F |` { A } ) E! y x ( F |` { A } ) y )
5	4	anbi2i 730	. 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 3203	. . . . . . . 8 |- y e. _V
7	6	brres 5402	. . . . . . 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 2490	. . . . 5 |- ( A e. dom F -> ( E! y x ( F |` { A } ) y <-> E! y ( x F y /\ x e. { A } ) ) )
10	9	ralbidv 2986	. . . 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 5439	. . . . 5 |- ( A e. dom F -> A e. dom ( F |` { A } ) )
12		eldmressn 41203	. . . . 5 |- ( x e. dom ( F |` { A } ) -> x = A )
13		breq1 4656	. . . . . . . 8 |- ( x = A -> ( x F y <-> A F y ) )
14	13	anbi1d 741	. . . . . . 7 |- ( x = A -> ( ( x F y /\ x e. { A } ) <-> ( A F y /\ x e. { A } ) ) )
15		velsn 4193	. . . . . . . . 9 |- ( x e. { A } <-> x = A )
16	15	biimpri 218	. . . . . . . 8 |- ( x = A -> x e. { A } )
17	16	biantrud 528	. . . . . . 7 |- ( x = A -> ( A F y <-> ( A F y /\ x e. { A } ) ) )
18	14, 17	bitr4d 271	. . . . . 6 |- ( x = A -> ( ( x F y /\ x e. { A } ) <-> A F y ) )
19	18	eubidv 2490	. . . . 5 |- ( x = A -> ( E! y ( x F y /\ x e. { A } ) <-> E! y A F y ) )
20	11, 12, 19	ralbinrald 41199	. . . 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 268	. . 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 669	. 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 286	1 |- ( F defAt A <-> ( A e. dom F /\ E! y A F y ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E!weu 2470   A.wral 2912   {csn 4177   class class class wbr 4653   dom cdm 5114   |` cres 5116   Rel wrel 5119   Fun wfun 5882   defAt wdfat 41193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-fun 5890  df-dfat 41196

This theorem is referenced by:  afveu  41233  rlimdmafv  41257