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Theorem 2ndval2 7940
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 5707 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 3450 . . . . . 6 𝑥 ∈ V
3 vex 3450 . . . . . 6 𝑦 ∈ V
42, 3op2nd 7931 . . . . 5 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
52, 3op2ndb 6180 . . . . 5 {⟨𝑥, 𝑦⟩} = 𝑦
64, 5eqtr4i 2768 . . . 4 (2nd ‘⟨𝑥, 𝑦⟩) = {⟨𝑥, 𝑦⟩}
7 fveq2 6843 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8 sneq 4597 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
98cnveqd 5832 . . . . . . 7 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
109inteqd 4913 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1110inteqd 4913 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1211inteqd 4913 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
136, 7, 123eqtr4a 2803 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
1413exlimivv 1936 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
151, 14sylbi 216 1 (𝐴 ∈ (V × V) → (2nd𝐴) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wex 1782  wcel 2107  Vcvv 3446  {csn 4587  cop 4593   cint 4908   × cxp 5632  ccnv 5633  cfv 6497  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-2nd 7923
This theorem is referenced by: (None)
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