Theorem 1stdm 6291

Description: The first ordered pair component of a member of a relation belongs to the domain of the relation. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
1stdm	|- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R )

Proof of Theorem 1stdm

Step	Hyp	Ref	Expression
1		df-rel 4700	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3201	. . 3 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		1stval2 6264	. . 3 |- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A )
5	3, 4	syl 14	. 2 |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) = |^| |^| A )
6		elreldm 4923	. 2 |- ( ( Rel R /\ A e. R ) -> |^| |^| A e. dom R )
7	5, 6	eqeltrd 2284	1 |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   |^|cint 3899   X. cxp 4691   dom cdm 4693   Rel wrel 4698   ` cfv 5290   1stc1st 6247

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-iota 5251  df-fun 5292  df-fv 5298  df-1st 6249

This theorem is referenced by: (None)