Theorem rnsnop 4924

Description: The range of a singleton of an ordered pair is the singleton of the second member. (Contributed by NM, 24-Jul-2004.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypothesis

Ref	Expression
cnvsn.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
rnsnop	⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}

Proof of Theorem rnsnop

Step	Hyp	Ref	Expression
1		cnvsn.1	. 2 ⊢ 𝐴 ∈ V
2		rnsnopg 4922	. 2 ⊢ (𝐴 ∈ V → ran {⟨𝐴, 𝐵⟩} = {𝐵})
3	1, 2	ax-mp 7	1 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}

Syntax hints:   = wceq 1290   ∈ wcel 1439  Vcvv 2620  {csn 3450  ⟨cop 3453  ran crn 4453

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4458  df-rel 4459  df-cnv 4460  df-dm 4462  df-rn 4463

This theorem is referenced by:  op2nda  4928  fpr  5493  en1  6570