Theorem op2nda 6087

Description: Extract the second member of an ordered pair. (See op1sta 6084 to extract the first member, op2ndb 6086 for an alternate version, and op2nd 7700 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 6083	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}
3	2	unieqi 4853	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = ∪ {𝐵}
4		cnvsn.2	. . 3 ⊢ 𝐵 ∈ V
5	4	unisn 4860	. 2 ⊢ ∪ {𝐵} = 𝐵
6	3, 5	eqtri 2846	1 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1537   ∈ wcel 2114  Vcvv 3496  {csn 4569  ⟨cop 4575  ∪ cuni 4840  ran crn 5558

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568

This theorem is referenced by:  elxp4  7629  elxp5  7630  op2nd  7700  fo2nd  7712  f2ndres  7716  ixpsnf1o  8504  xpassen  8613  xpdom2  8614