Theorem 1stdm 5049

Description: The first ordered pair component of a member of a relation belongs to the domain of the relation.

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 4001	. . . . . 6 |- (Rel R <-> R C_ (_V X. _V))
2	1	biimpi 168	. . . . 5 |- (Rel R -> R C_ (_V X. _V))
3	2	sseld 2619	. . . 4 |- (Rel R -> (A e. R -> A e. (_V X. _V)))
4	3	imp 377	. . 3 |- ((Rel R /\ A e. R) -> A e. (_V X. _V))
5		1stval2 5030	. . 3 |- (A e. (_V X. _V) -> (1st` A) = |^||^|A)
6	4, 5	syl 12	. 2 |- ((Rel R /\ A e. R) -> (1st` A) = |^||^|A)
7		elreldm 4185	. 2 |- ((Rel R /\ A e. R) -> |^||^|A e. dom R)
8	6, 7	eqeltrd 1971	1 |- ((Rel R /\ A e. R) -> (1st` A) e. dom R)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  |^|cint 3214   X. cxp 3984  dom cdm 3986  Rel wrel 3991  ` cfv 3998  1stc1st 5018

This theorem is referenced by:  frxp 13951  11st22nd 14348  issubcat 15193

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020

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