Theorem elreldm 4723

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 4504	. . . . 5 |- ( Rel A <-> A C_ ( _V X. _V ) )
2		ssel 3055	. . . . 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 4559	. . . 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 2175	. . . . . 6 |- ( B = <. x , y >. -> ( B e. A <-> <. x , y >. e. A ) )
7		vex 2658	. . . . . . 7 |- x e. _V
8		vex 2658	. . . . . . 7 |- y e. _V
9	7, 8	opeldm 4700	. . . . . 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 3738	. . . . . . . 8 |- ( B = <. x , y >. -> |^| B = |^| <. x , y >. )
12	11	inteqd 3740	. . . . . . 7 |- ( B = <. x , y >. -> |^| |^| B = |^| |^| <. x , y >. )
13	7, 8	op1stb 4357	. . . . . . 7 |- |^| |^| <. x , y >. = x
14	12, 13	syl6eq 2161	. . . . . 6 |- ( B = <. x , y >. -> |^| |^| B = x )
15	14	eleq1d 2181	. . . . 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 1848	. . 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 1312   E.wex 1449   e. wcel 1461   _Vcvv 2655   C_ wss 3035   <.cop 3494   |^|cint 3735   X. cxp 4495   dom cdm 4497   Rel wrel 4502

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089

This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-v 2657  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-int 3736  df-br 3894  df-opab 3948  df-xp 4503  df-rel 4504  df-dm 4507

This theorem is referenced by:  1stdm  6032  fundmen  6652