Theorem elreldm 3335

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

Assertion

Ref	Expression
elreldm	|- ((Rel A /\ B e. A) -> |^||^|B e. dom A)

Proof of Theorem elreldm

Step	Hyp	Ref	Expression
1		df-rel 3182	. . . . 5 |- (Rel A <-> A (_ (V X. V))
2		ssel 2061	. . . . 5 |- (A (_ (V X. V) -> (B e. A -> B e. (V X. V)))
3	1, 2	sylbi 199	. . . 4 |- (Rel A -> (B e. A -> B e. (V X. V)))
4		elvv 3225	. . . 4 |- (B e. (V X. V) <-> E.xE.y B = <.x, y>.)
5	3, 4	syl6ib 212	. . 3 |- (Rel A -> (B e. A -> E.xE.y B = <.x, y>.))
6		eleq1 1533	. . . . . 6 |- (B = <.x, y>. -> (B e. A <-> <.x, y>. e. A))
7		visset 1811	. . . . . . 7 |- x e. V
8	7	opeldm 3311	. . . . . 6 |- (<.x, y>. e. A -> x e. dom A)
9	6, 8	syl6bi 214	. . . . 5 |- (B = <.x, y>. -> (B e. A -> x e. dom A))
10		inteq 2533	. . . . . . . 8 |- (B = <.x, y>. -> |^|B = |^|<.x, y>.)
11	10	inteqd 2535	. . . . . . 7 |- (B = <.x, y>. -> |^||^|B = |^||^|<.x, y>.)
12	7	op1stb 2910	. . . . . . 7 |- |^||^|<.x, y>. = x
13	11, 12	syl6eq 1522	. . . . . 6 |- (B = <.x, y>. -> |^||^|B = x)
14	13	eleq1d 1539	. . . . 5 |- (B = <.x, y>. -> (|^||^|B e. dom A <-> x e. dom A))
15	9, 14	sylibrd 204	. . . 4 |- (B = <.x, y>. -> (B e. A -> |^||^|B e. dom A))
16	15	19.23aivv 1296	. . 3 |- (E.xE.y B = <.x, y>. -> (B e. A -> |^||^|B e. dom A))
17	5, 16	syli 54	. 2 |- (Rel A -> (B e. A -> |^||^|B e. dom A))
18	17	imp 350	1 |- ((Rel A /\ B e. A) -> |^||^|B e. dom A)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  E.wex 979  Vcvv 1809   (_ wss 2045  <.cop 2409  |^|cint 2530   X. cxp 3165  dom cdm 3167  Rel wrel 3172

This theorem is referenced by:  1stdm 4106  fundmen 4422

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-int 2531  df-br 2617  df-opab 2664  df-xp 3181  df-rel 3182  df-dm 3185

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