Theorem 1stdm 6376

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 4756	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3238	. . 3 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		1stval2 6349	. . 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 4983	. 2 |- ( ( Rel R /\ A e. R ) -> |^| |^| A e. dom R )
7	5, 6	eqeltrd 2309	1 |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   _Vcvv 2813   C_ wss 3211   |^|cint 3949   X. cxp 4747   dom cdm 4749   Rel wrel 4754   ` cfv 5352   1stc1st 6332

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-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fv 5360  df-1st 6334

This theorem is referenced by: (None)