Theorem elreldm 4964

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 4738	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
2		ssel 3222	. . . . 5 |- ( A C_ ( _V X. _V ) -> ( B e. A -> B e. ( _V X. _V ) ) )
3	1, 2	sylbi 121	. . . 4 |- ( Rel A -> ( B e. A -> B e. ( _V X. _V ) ) )
4		elvv 4794	. . . 4 |- ( B e. ( _V X. _V ) <-> E. x E. y B = <. x , y >. )
5	3, 4	imbitrdi 161	. . 3 |- ( Rel A -> ( B e. A -> E. x E. y B = <. x , y >. ) )
6		eleq1 2294	. . . . . 6 |- ( B = <. x , y >. -> ( B e. A <-> <. x , y >. e. A ) )
7		vex 2806	. . . . . . 7 |- x e. _V
8		vex 2806	. . . . . . 7 |- y e. _V
9	7, 8	opeldm 4940	. . . . . 6 |- ( <. x , y >. e. A -> x e. dom A )
10	6, 9	biimtrdi 163	. . . . 5 |- ( B = <. x , y >. -> ( B e. A -> x e. dom A ) )
11		inteq 3936	. . . . . . . 8 |- ( B = <. x , y >. -> |^| B = |^| <. x , y >. )
12	11	inteqd 3938	. . . . . . 7 |- ( B = <. x , y >. -> |^| |^| B = |^| |^| <. x , y >. )
13	7, 8	op1stb 4581	. . . . . . 7 |- |^| |^| <. x , y >. = x
14	12, 13	eqtrdi 2280	. . . . . 6 |- ( B = <. x , y >. -> |^| |^| B = x )
15	14	eleq1d 2300	. . . . 5 |- ( B = <. x , y >. -> ( |^| |^| B e. dom A <-> x e. dom A ) )
16	10, 15	sylibrd 169	. . . 4 |- ( B = <. x , y >. -> ( B e. A -> |^| |^| B e. dom A ) )
17	16	exlimivv 1945	. . 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 124	1 |- ( ( Rel A /\ B e. A ) -> |^| |^| B e. dom A )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   C_ wss 3201   <.cop 3676   |^|cint 3933   X. cxp 4729   dom cdm 4731   Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-int 3934  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-dm 4741

This theorem is referenced by:  1stdm  6354  fundmen  7024