Theorem 1stdm 6328

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 4726	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3224	. . 3 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		1stval2 6301	. . 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 4950	. 2 |- ( ( Rel R /\ A e. R ) -> |^| |^| A e. dom R )
7	5, 6	eqeltrd 2306	1 |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   C_ wss 3197   |^|cint 3923   X. cxp 4717   dom cdm 4719   Rel wrel 4724   ` cfv 5318   1stc1st 6284

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fv 5326  df-1st 6286

This theorem is referenced by: (None)