Theorem elreldm 4837

Description: The first member of an ordered pair in a relation belongs to the domain of the relation. (Contributed by NM, 28-Jul-2004.)

Assertion

Ref	Expression
elreldm	|- ( ( Rel A /\ B e. A ) -> |^| |^| B e. dom A )

Proof of Theorem elreldm

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rel 4618	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
2		ssel 3141	. . . . 5 |- ( A C_ ( _V X. _V ) -> ( B e. A -> B e. ( _V X. _V ) ) )
3	1, 2	sylbi 120	. . . 4 |- ( Rel A -> ( B e. A -> B e. ( _V X. _V ) ) )
4		elvv 4673	. . . 4 |- ( B e. ( _V X. _V ) <-> E. x E. y B = <. x , y >. )
5	3, 4	syl6ib 160	. . 3 |- ( Rel A -> ( B e. A -> E. x E. y B = <. x , y >. ) )
6		eleq1 2233	. . . . . 6 |- ( B = <. x , y >. -> ( B e. A <-> <. x , y >. e. A ) )
7		vex 2733	. . . . . . 7 |- x e. _V
8		vex 2733	. . . . . . 7 |- y e. _V
9	7, 8	opeldm 4814	. . . . . 6 |- ( <. x , y >. e. A -> x e. dom A )
10	6, 9	syl6bi 162	. . . . 5 |- ( B = <. x , y >. -> ( B e. A -> x e. dom A ) )
11		inteq 3834	. . . . . . . 8 |- ( B = <. x , y >. -> |^| B = |^| <. x , y >. )
12	11	inteqd 3836	. . . . . . 7 |- ( B = <. x , y >. -> |^| |^| B = |^| |^| <. x , y >. )
13	7, 8	op1stb 4463	. . . . . . 7 |- |^| |^| <. x , y >. = x
14	12, 13	eqtrdi 2219	. . . . . 6 |- ( B = <. x , y >. -> |^| |^| B = x )
15	14	eleq1d 2239	. . . . 5 |- ( B = <. x , y >. -> ( |^| |^| B e. dom A <-> x e. dom A ) )
16	10, 15	sylibrd 168	. . . 4 |- ( B = <. x , y >. -> ( B e. A -> |^| |^| B e. dom A ) )
17	16	exlimivv 1889	. . 3 |- ( E. x E. y B = <. x , y >. -> ( B e. A -> |^| |^| B e. dom A ) )
18	5, 17	syli 37	. 2 |- ( Rel A -> ( B e. A -> |^| |^| B e. dom A ) )
19	18	imp 123	1 |- ( ( Rel A /\ B e. A ) -> |^| |^| B e. dom A )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   _Vcvv 2730   C_ wss 3121   <.cop 3586   |^|cint 3831   X. cxp 4609   dom cdm 4611   Rel wrel 4616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-int 3832  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-dm 4621

This theorem is referenced by:  1stdm  6161  fundmen  6784