Theorem brelrn 4895

Description: The second argument of a binary relation belongs to its range. (Contributed by NM, 13-Aug-2004.)

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.1	. 2 ⊢ 𝐴 ∈ V
2		brelrn.2	. 2 ⊢ 𝐵 ∈ V
3		brelrng 4893	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐴𝐶𝐵) → 𝐵 ∈ ran 𝐶)
4	1, 2, 3	mp3an12 1338	1 ⊢ (𝐴𝐶𝐵 → 𝐵 ∈ ran 𝐶)

Syntax hints:   → wi 4   ∈ wcel 2164  Vcvv 2760   class class class wbr 4029  ran crn 4660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670

This theorem is referenced by:  opelrn  4896  dfco2a  5166  cores  5169  dffun9  5283  funcnv  5315  rntpos  6310  tfrexlem  6387