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Theorem 2ndval2 5725
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 4345 . 2 (A (V × V) ↔ xy A = ⟨x, y⟩)
2 vex 2554 . . . . . 6 x V
3 vex 2554 . . . . . 6 y V
42, 3op2nd 5716 . . . . 5 (2nd ‘⟨x, y⟩) = y
52, 3op2ndb 4747 . . . . 5 {⟨x, y⟩} = y
64, 5eqtr4i 2060 . . . 4 (2nd ‘⟨x, y⟩) = {⟨x, y⟩}
7 fveq2 5121 . . . 4 (A = ⟨x, y⟩ → (2ndA) = (2nd ‘⟨x, y⟩))
8 sneq 3378 . . . . . . . 8 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
98cnveqd 4454 . . . . . . 7 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
109inteqd 3611 . . . . . 6 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
1110inteqd 3611 . . . . 5 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
1211inteqd 3611 . . . 4 (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
136, 7, 123eqtr4a 2095 . . 3 (A = ⟨x, y⟩ → (2ndA) = {A})
1413exlimivv 1773 . 2 (xy A = ⟨x, y⟩ → (2ndA) = {A})
151, 14sylbi 114 1 (A (V × V) → (2ndA) = {A})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   cint 3606   × cxp 4286  ccnv 4287  cfv 4845  2nd c2nd 5708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-2nd 5710
This theorem is referenced by: (None)
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