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Theorem dmsnopg 4735
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
dmsnopg (B 𝑉 → dom {⟨A, B⟩} = {A})

Proof of Theorem dmsnopg
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . 6 x V
2 vex 2554 . . . . . 6 y V
31, 2opth1 3964 . . . . 5 (⟨x, y⟩ = ⟨A, B⟩ → x = A)
43exlimiv 1486 . . . 4 (yx, y⟩ = ⟨A, B⟩ → x = A)
5 opeq1 3540 . . . . 5 (x = A → ⟨x, B⟩ = ⟨A, B⟩)
6 opeq2 3541 . . . . . . 7 (y = B → ⟨x, y⟩ = ⟨x, B⟩)
76eqeq1d 2045 . . . . . 6 (y = B → (⟨x, y⟩ = ⟨A, B⟩ ↔ ⟨x, B⟩ = ⟨A, B⟩))
87spcegv 2635 . . . . 5 (B 𝑉 → (⟨x, B⟩ = ⟨A, B⟩ → yx, y⟩ = ⟨A, B⟩))
95, 8syl5 28 . . . 4 (B 𝑉 → (x = Ayx, y⟩ = ⟨A, B⟩))
104, 9impbid2 131 . . 3 (B 𝑉 → (yx, y⟩ = ⟨A, B⟩ ↔ x = A))
111eldm2 4476 . . . 4 (x dom {⟨A, B⟩} ↔ yx, y {⟨A, B⟩})
121, 2opex 3957 . . . . . 6 x, y V
1312elsnc 3390 . . . . 5 (⟨x, y {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
1413exbii 1493 . . . 4 (yx, y {⟨A, B⟩} ↔ yx, y⟩ = ⟨A, B⟩)
1511, 14bitri 173 . . 3 (x dom {⟨A, B⟩} ↔ yx, y⟩ = ⟨A, B⟩)
16 elsn 3382 . . 3 (x {A} ↔ x = A)
1710, 15, 163bitr4g 212 . 2 (B 𝑉 → (x dom {⟨A, B⟩} ↔ x {A}))
1817eqrdv 2035 1 (B 𝑉 → dom {⟨A, B⟩} = {A})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wex 1378   wcel 1390  {csn 3367  cop 3370  dom cdm 4288
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-bndl 1396  ax-4 1397  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
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 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-dm 4298
This theorem is referenced by:  dmpropg  4736  dmsnop  4737  rnsnopg  4742  elxp4  4751  fnsng  4890  funprg  4892  funtpg  4893  fntpg  4898
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