Theorem 1stdm 6124

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 4590	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 119	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3128	. . 3 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		1stval2 6097	. . 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 4809	. 2 |- ( ( Rel R /\ A e. R ) -> |^| |^| A e. dom R )
7	5, 6	eqeltrd 2234	1 |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1335   e. wcel 2128   _Vcvv 2712   C_ wss 3102   |^|cint 3807   X. cxp 4581   dom cdm 4583   Rel wrel 4588   ` cfv 5167   1stc1st 6080

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4134  ax-pr 4168  ax-un 4392

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-iota 5132  df-fun 5169  df-fv 5175  df-1st 6082

This theorem is referenced by: (None)