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Theorem 2ndval2 6054
 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 4601 . 2 (𝐴 ∈ (V × V) ↔ ∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩)
2 vex 2689 . . . . . 6 𝑥 ∈ V
3 vex 2689 . . . . . 6 𝑦 ∈ V
42, 3op2nd 6045 . . . . 5 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
52, 3op2ndb 5022 . . . . 5 {⟨𝑥, 𝑦⟩} = 𝑦
64, 5eqtr4i 2163 . . . 4 (2nd ‘⟨𝑥, 𝑦⟩) = {⟨𝑥, 𝑦⟩}
7 fveq2 5421 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = (2nd ‘⟨𝑥, 𝑦⟩))
8 sneq 3538 . . . . . . . 8 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
98cnveqd 4715 . . . . . . 7 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
109inteqd 3776 . . . . . 6 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1110inteqd 3776 . . . . 5 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
1211inteqd 3776 . . . 4 (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
136, 7, 123eqtr4a 2198 . . 3 (𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
1413exlimivv 1868 . 2 (∃𝑥𝑦 𝐴 = ⟨𝑥, 𝑦⟩ → (2nd𝐴) = {𝐴})
151, 14sylbi 120 1 (𝐴 ∈ (V × V) → (2nd𝐴) = {𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Vcvv 2686  {csn 3527  ⟨cop 3530  ∩ cint 3771   × cxp 4537  ◡ccnv 4538  ‘cfv 5123  2nd c2nd 6037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-2nd 6039 This theorem is referenced by: (None)
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