Theorem brelrn 3351

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

Hypotheses

Ref	Expression
brelrn.1	⊢ A ∈ V
brelrn.2	⊢ B ∈ V

Assertion

Ref	Expression
brelrn	⊢ (ACB → B ∈ ran C)

Proof of Theorem brelrn

Step	Hyp	Ref	Expression
1		brelrn.2	. 2 ⊢ B ∈ V
2		brelrn.1	. . 3 ⊢ A ∈ V
3		brelrng 3350	. . 3 ⊢ ((A ∈ V ⋀ B ∈ V ⋀ ACB) → B ∈ ran C)
4	2, 3	mp3an1 905	. 2 ⊢ ((B ∈ V ⋀ ACB) → B ∈ ran C)
5	1, 4	mpan 697	1 ⊢ (ACB → B ∈ ran C)

Syntax hints:   → wi 3   ∈ wcel 960  Vcvv 1814   class class class wbr 2625  ran crn 3178

This theorem is referenced by:  opelrn 3352  cores 3506  dffun8 3548  funcnv 3564  cbvfo 3892  psdmrn 8651  rnhmpha 10529

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 2709  ax-pow 2749  ax-pr 2786

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 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-br 2626  df-opab 2673  df-cnv 3193  df-dm 3195  df-rn 3196

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