Theorem op2nda 6120

Description: Extract the second member of an ordered pair. (See op1sta 6117 to extract the first member, op2ndb 6119 for an alternate version, and op2nd 7813 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 6116	. . 3 ⊢ ran {⟨𝐴, 𝐵⟩} = {𝐵}
3	2	unieqi 4849	. 2 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = ∪ {𝐵}
4		cnvsn.2	. . 3 ⊢ 𝐵 ∈ V
5	4	unisn 4858	. 2 ⊢ ∪ {𝐵} = 𝐵
6	3, 5	eqtri 2766	1 ⊢ ∪ ran {⟨𝐴, 𝐵⟩} = 𝐵

Syntax hints:   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  ⟨cop 4564  ∪ cuni 4836  ran crn 5581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591

This theorem is referenced by:  elxp4  7743  elxp5  7744  op2nd  7813  fo2nd  7825  f2ndres  7829  ixpsnf1o  8684  xpassen  8806  xpdom2  8807