Theorem elreldm 4830

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 4611	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
2		ssel 3136	. . . . 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 4666	. . . 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 2229	. . . . . 6 |- ( B = <. x , y >. -> ( B e. A <-> <. x , y >. e. A ) )
7		vex 2729	. . . . . . 7 |- x e. _V
8		vex 2729	. . . . . . 7 |- y e. _V
9	7, 8	opeldm 4807	. . . . . 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 3827	. . . . . . . 8 |- ( B = <. x , y >. -> |^| B = |^| <. x , y >. )
12	11	inteqd 3829	. . . . . . 7 |- ( B = <. x , y >. -> |^| |^| B = |^| |^| <. x , y >. )
13	7, 8	op1stb 4456	. . . . . . 7 |- |^| |^| <. x , y >. = x
14	12, 13	eqtrdi 2215	. . . . . 6 |- ( B = <. x , y >. -> |^| |^| B = x )
15	14	eleq1d 2235	. . . . 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 1884	. . 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 1343   E.wex 1480   e. wcel 2136   _Vcvv 2726   C_ wss 3116   <.cop 3579   |^|cint 3824   X. cxp 4602   dom cdm 4604   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-int 3825  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-dm 4614

This theorem is referenced by:  1stdm  6150  fundmen  6772