MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  2ndval2 Structured version   Visualization version   GIF version

Theorem 2ndval2 8031
Description: Alternate value of the function that extracts the second member of an ordered pair. Definition 5.13 (ii) of [Monk1] p. 52. (Contributed by NM, 18-Aug-2006.)
Assertion
Ref Expression
2ndval2 (𝐴 ∈ (V × V) → (2nd𝐴) = {𝐴})

Proof of Theorem 2ndval2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elvv 5763 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3482 . . . . . 6 𝑥 ∈ V
3 vex 3482 . . . . . 6 𝑦 ∈ V
42, 3op2nd 8022 . . . . 5 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
52, 3op2ndb 6249 . . . . 5 {⟨𝑥, 𝑦⟩} = 𝑦
64, 5eqtr4i 2766 . . . 4 (2nd ‘⟨𝑥, 𝑦⟩) = {⟨𝑥, 𝑦⟩}
7 fveq2 6907 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8 sneq 4641 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
98cnveqd 5889 . . . . . . 7 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
109inteqd 4956 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1110inteqd 4956 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1211inteqd 4956 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
136, 7, 123eqtr4a 2801 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
1413exlimivv 1930 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
151, 14sylbi 217 1 (𝐴 ∈ (V × V) → (2nd𝐴) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  {csn 4631  cop 4637   cint 4951   × cxp 5687  ccnv 5688  cfv 6563  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fv 6571  df-2nd 8014
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator