Theorem rnsnop 5014

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 5012	. 2 ⊢ (𝐴 ∈ V → ran {⟨𝐴, 𝐵⟩} = {𝐵})
3	1, 2	ax-mp 5	1 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}

Syntax hints:   = wceq 1331   ∈ wcel 1480  Vcvv 2681  {csn 3522  ⟨cop 3525  ran crn 4535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545

This theorem is referenced by:  op2nda  5018  fpr  5595  en1  6686