Theorem rnsnop 6182

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 6179	. 2 ⊢ (𝐴 ∈ V → ran {⟨𝐴, 𝐵⟩} = {𝐵})
3	1, 2	ax-mp 5	1 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3440  {csn 4580  ⟨cop 4586  ran crn 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  op2nda  6186  fpr  7099  en1  8963  fodomfi  9214  fodomfiOLD  9232  dcomex  10359  s1rn  14525  axlowdimlem13  29029  ex-rn  30517  ex-ima  30519  ptrest  37822  poimirlem3  37826  gidsn  38155  zrdivrng  38156