Theorem rnsnop 5268

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 5266	. 2 ⊢ (𝐴 ∈ V → ran {⟨𝐴, 𝐵⟩} = {𝐵})
3	1, 2	ax-mp 5	1 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}

This proof depends on syntax axioms:   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3709  ⟨cop 3712  ran crn 4775

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-dm 4784  df-rn 4785

This theorem is used by:  op2nda  5272  fpr  5897  en1  7086  s1rn  11386