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

Theorem 2ndval2 7696
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 5619 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3495 . . . . . 6 𝑥 ∈ V
3 vex 3495 . . . . . 6 𝑦 ∈ V
42, 3op2nd 7687 . . . . 5 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
52, 3op2ndb 6077 . . . . 5 {⟨𝑥, 𝑦⟩} = 𝑦
64, 5eqtr4i 2844 . . . 4 (2nd ‘⟨𝑥, 𝑦⟩) = {⟨𝑥, 𝑦⟩}
7 fveq2 6663 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8 sneq 4567 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
98cnveqd 5739 . . . . . . 7 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
109inteqd 4872 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1110inteqd 4872 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1211inteqd 4872 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
136, 7, 123eqtr4a 2879 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
1413exlimivv 1924 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
151, 14sylbi 218 1 (𝐴 ∈ (V × V) → (2nd𝐴) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  {csn 4557  cop 4563   cint 4867   × cxp 5546  ccnv 5547  cfv 6348  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-2nd 7679
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator