Theorem op2nda 6248

Description: Extract the second member of an ordered pair. (See op1sta 6245 to extract the first member, op2ndb 6247 for an alternate version, and op2nd 8023 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 6244	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}
3	2	unieqi 4919	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = ∪ {𝐵}
4		cnvsn.2	. . 3 ⊢ 𝐵 ∈ V
5	4	unisn 4926	. 2 ⊢ ∪ {𝐵} = 𝐵
6	3, 5	eqtri 2765	1 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1540   ∈ wcel 2108  Vcvv 3480  {csn 4626  ⟨cop 4632  ∪ cuni 4907  ran crn 5686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696

This theorem is referenced by:  elxp4  7944  elxp5  7945  op2nd  8023  fo2nd  8035  f2ndres  8039  ixpsnf1o  8978  xpassen  9106  xpdom2  9107