MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op2nda Structured version   Visualization version   GIF version

Theorem op2nda 5808
Description: Extract the second member of an ordered pair. (See op1sta 5804 to extract the first member, op2ndb 5807 for an alternate version, and op2nd 7379 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
StepHypRef Expression
1 cnvsn.1 . . . 4 𝐴 ∈ V
21rnsnop 5803 . . 3 ran {⟨𝐴, 𝐵⟩} = {𝐵}
32unieqi 4605 . 2 ran {⟨𝐴, 𝐵⟩} = {𝐵}
4 cnvsn.2 . . 3 𝐵 ∈ V
54unisn 4612 . 2 {𝐵} = 𝐵
63, 5eqtri 2787 1 ran {⟨𝐴, 𝐵⟩} = 𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  wcel 2155  Vcvv 3350  {csn 4336  cop 4342   cuni 4596  ran crn 5280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-xp 5285  df-rel 5286  df-cnv 5287  df-dm 5289  df-rn 5290
This theorem is referenced by:  elxp4  7312  elxp5  7313  op2nd  7379  fo2nd  7391  f2ndres  7395  ixpsnf1o  8157  xpassen  8265  xpdom2  8266
  Copyright terms: Public domain W3C validator