Theorem 1stdm 7215

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 5121	. . . . 5 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 206	. . . 4 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3603	. . 3 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		1stval2 7185	. . 3 |- ( A e. ( _V X. _V ) -> ( 1st ` A ) = |^| |^| A )
5	3, 4	syl 17	. 2 |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) = |^| |^| A )
6		elreldm 5350	. 2 |- ( ( Rel R /\ A e. R ) -> |^| |^| A e. dom R )
7	5, 6	eqeltrd 2701	1 |- ( ( Rel R /\ A e. R ) -> ( 1st ` A ) e. dom R )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   |^|cint 4475   X. cxp 5112   dom cdm 5114   Rel wrel 5119   ` cfv 5888   1stc1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168

This theorem is referenced by:  frxp  7287  dprd2dlem2  18439  dprd2da  18441  gsummpt2d  29781