Theorem aoprssdm 41282

Description: Domain of closure of an operation. In contrast to oprssdm 6815, no additional property for S ( -. (/) e. S) is required! (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

Ref	Expression
aoprssdm.1	|- ( ( x e. S /\ y e. S ) -> (( x F y)) e. S )

Assertion

Ref	Expression
aoprssdm	|- ( S X. S ) C_ dom F

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

Proof of Theorem aoprssdm

Step	Hyp	Ref	Expression
1		relxp 5227	. 2 |- Rel ( S X. S )
2		opelxp 5146	. . 3 |- ( <. x , y >. e. ( S X. S ) <-> ( x e. S /\ y e. S ) )
3		df-aov 41198	. . . . 5 |- (( x F y)) = ( F''' <. x , y >. )
4		aoprssdm.1	. . . . 5 |- ( ( x e. S /\ y e. S ) -> (( x F y)) e. S )
5	3, 4	syl5eqelr 2706	. . . 4 |- ( ( x e. S /\ y e. S ) -> ( F''' <. x , y >. ) e. S )
6		afvvdm 41221	. . . 4 |- ( ( F''' <. x , y >. ) e. S -> <. x , y >. e. dom F )
7	5, 6	syl 17	. . 3 |- ( ( x e. S /\ y e. S ) -> <. x , y >. e. dom F )
8	2, 7	sylbi 207	. 2 |- ( <. x , y >. e. ( S X. S ) -> <. x , y >. e. dom F )
9	1, 8	relssi 5211	1 |- ( S X. S ) C_ dom F

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   <.cop 4183   X. cxp 5112   dom cdm 5114  '''cafv 41194   ((caov 41195

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-8 1992  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-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-opab 4713  df-xp 5120  df-rel 5121  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)