Theorem 1stdm 4115

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 3191	. . . . . 6 |- (Rel R <-> R (_ (V X. V))
2	1	biimp 151	. . . . 5 |- (Rel R -> R (_ (V X. V))
3	2	sseld 2070	. . . 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 4095	. . 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 3344	. 2 |- ((Rel R /\ A e. R) -> |^||^|A e. dom R)
8	6, 7	eqeltrd 1551	1 |- ((Rel R /\ A e. R) -> (1st` A) e. dom R)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   (_ wss 2050  |^|cint 2537   X. cxp 3174  dom cdm 3176  Rel wrel 3181  ` cfv 3188  1stc1st 4083

This theorem is referenced by:  11st22nd 10448

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-1st 4085

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