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Theorem dmsnm 4728
Description: The domain of a singleton is inhabited iff the singleton argument is an ordered pair. (Contributed by Jim Kingdon, 15-Dec-2018.)
Assertion
Ref Expression
dmsnm (A (V × V) ↔ x x dom {A})
Distinct variable group:   x,A

Proof of Theorem dmsnm
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elvv 4344 . 2 (A (V × V) ↔ xy A = ⟨x, y⟩)
2 vex 2554 . . . . 5 x V
32eldm 4474 . . . 4 (x dom {A} ↔ y x{A}y)
4 df-br 3755 . . . . . 6 (x{A}y ↔ ⟨x, y {A})
5 vex 2554 . . . . . . . 8 y V
62, 5opex 3956 . . . . . . 7 x, y V
76elsnc 3389 . . . . . 6 (⟨x, y {A} ↔ ⟨x, y⟩ = A)
8 eqcom 2039 . . . . . 6 (⟨x, y⟩ = AA = ⟨x, y⟩)
94, 7, 83bitri 195 . . . . 5 (x{A}yA = ⟨x, y⟩)
109exbii 1493 . . . 4 (y x{A}yy A = ⟨x, y⟩)
113, 10bitr2i 174 . . 3 (y A = ⟨x, y⟩ ↔ x dom {A})
1211exbii 1493 . 2 (xy A = ⟨x, y⟩ ↔ x x dom {A})
131, 12bitri 173 1 (A (V × V) ↔ x x dom {A})
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  {csn 3366  cop 3369   class class class wbr 3754   × cxp 4285  dom cdm 4287
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-xp 4293  df-dm 4297
This theorem is referenced by:  rnsnm  4729  dmsn0  4730  dmsn0el  4732  relsn2m  4733
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