Theorem op2nda 6186

Description: Extract the second member of an ordered pair. (See op1sta 6183 to extract the first member, op2ndb 6185 for an alternate version, and op2nd 7942 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 6182	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}
3	2	unieqi 4875	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = ∪ {𝐵}
4		cnvsn.2	. . 3 ⊢ 𝐵 ∈ V
5	4	unisn 4882	. 2 ⊢ ∪ {𝐵} = 𝐵
6	3, 5	eqtri 2759	1 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3440  {csn 4580  ⟨cop 4586  ∪ cuni 4863  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-uni 4864  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:  elxp4  7864  elxp5  7865  op2nd  7942  fo2nd  7954  f2ndres  7958  ixpsnf1o  8876  xpassen  8999  xpdom2  9000