Theorem 1stdm 4093

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 3175	. . . . . 6 |- (Rel R <-> R (_ (V X. V))
2	1	biimp 151	. . . . 5 |- (Rel R -> R (_ (V X. V))
3	2	sseld 2057	. . . 4 |- (Rel R -> (A e. R -> A e. (V X. V)))
4	3	imp 350	. . 3 |- ((Rel R /\ A e. R) -> A e. (V X. V))
5		1stval2 4073	. . 3 |- (A e. (V X. V) -> (1st` A) = |^||^|A)
6	4, 5	syl 10	. 2 |- ((Rel R /\ A e. R) -> (1st` A) = |^||^|A)
7		elreldm 3327	. 2 |- ((Rel R /\ A e. R) -> |^||^|A e. dom R)
8	6, 7	eqeltrd 1540	1 |- ((Rel R /\ A e. R) -> (1st` A) e. dom R)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802   (_ wss 2037  |^|cint 2523   X. cxp 3158  dom cdm 3160  Rel wrel 3165  ` cfv 3172  1stc1st 4061

This theorem is referenced by:  11st22nd 10354

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-1st 4063

Copyright terms: Public domain