Theorem op2nda 6227

Description: Extract the second member of an ordered pair. (See op1sta 6224 to extract the first member, op2ndb 6226 for an alternate version, and op2nd 7991 for the preferred version.) (Contributed by NM, 17-Feb-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Hypotheses

Ref	Expression
cnvsn.1	⊢ 𝐴 ∈ V
cnvsn.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
op2nda	⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵

Proof of Theorem op2nda

Step	Hyp	Ref	Expression
1		cnvsn.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	rnsnop 6223	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}
3	2	unieqi 4885	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = ∪ {𝐵}
4		cnvsn.2	. . 3 ⊢ 𝐵 ∈ V
5	4	unisn 4892	. 2 ⊢ ∪ {𝐵} = 𝐵
6	3, 5	eqtri 2792	1 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1567   ∈ wcel 2149  Vcvv 3463  {csn 4591  ⟨cop 4597  ∪ cuni 4873  ran crn 5660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5665  df-rel 5666  df-cnv 5667  df-dm 5669  df-rn 5670

This theorem is referenced by:  elxp4  7915  elxp5  7916  op2nd  7991  fo2nd  8003  f2ndres  8007  ixpsnf1o  8932  xpassen  9055  xpdom2  9056