Theorem relelrng 3353

Description: The second argument of a binary relation belongs to its range.

Assertion

Ref	Expression
relelrng	⊢ ((B ∈ C ⋀ Rel R ⋀ ARB) → B ∈ ran R)

Proof of Theorem relelrng

Step	Hyp	Ref	Expression
1		brelrng 3349	. . 3 ⊢ ((A ∈ V ⋀ B ∈ C ⋀ ARB) → B ∈ ran R)
2	1	3com12 839	. 2 ⊢ ((B ∈ C ⋀ A ∈ V ⋀ ARB) → B ∈ ran R)
3		brrelex 3213	. . 3 ⊢ ((Rel R ⋀ ARB) → A ∈ V)
4	3	3adant1 799	. 2 ⊢ ((B ∈ C ⋀ Rel R ⋀ ARB) → A ∈ V)
5	2, 4	syld3an2 874	1 ⊢ ((B ∈ C ⋀ Rel R ⋀ ARB) → B ∈ ran R)

Syntax hints:   → wi 3   ⋀ w3a 777   ∈ wcel 960  Vcvv 1814   class class class wbr 2624  ran crn 3177  Rel wrel 3181

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-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-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  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-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-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195

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