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Theorem dmsnopg 4719
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 2538 . . . . . 6 x V
2 vex 2538 . . . . . 6 y V
31, 2opth1 3947 . . . . 5 (⟨x, y⟩ = ⟨A, B⟩ → x = A)
43exlimiv 1471 . . . 4 (yx, y⟩ = ⟨A, B⟩ → x = A)
5 opeq1 3523 . . . . 5 (x = A → ⟨x, B⟩ = ⟨A, B⟩)
6 opeq2 3524 . . . . . . 7 (y = B → ⟨x, y⟩ = ⟨x, B⟩)
76eqeq1d 2030 . . . . . 6 (y = B → (⟨x, y⟩ = ⟨A, B⟩ ↔ ⟨x, B⟩ = ⟨A, B⟩))
87spcegv 2618 . . . . 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 4460 . . . 4 (x dom {⟨A, B⟩} ↔ yx, y {⟨A, B⟩})
121, 2opex 3940 . . . . . 6 x, y V
1312elsnc 3373 . . . . 5 (⟨x, y {⟨A, B⟩} ↔ ⟨x, y⟩ = ⟨A, B⟩)
1413exbii 1478 . . . 4 (yx, y {⟨A, B⟩} ↔ yx, y⟩ = ⟨A, B⟩)
1511, 14bitri 173 . . 3 (x dom {⟨A, B⟩} ↔ yx, y⟩ = ⟨A, B⟩)
16 elsn 3365 . . 3 (x {A} ↔ x = A)
1710, 15, 163bitr4g 212 . 2 (B 𝑉 → (x dom {⟨A, B⟩} ↔ x {A}))
1817eqrdv 2020 1 (B 𝑉 → dom {⟨A, B⟩} = {A})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228  wex 1362   wcel 1374  {csn 3350  cop 3353  dom cdm 4272
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-dm 4282
This theorem is referenced by:  dmpropg  4720  dmsnop  4721  rnsnopg  4726  elxp4  4735  fnsng  4873  funprg  4875  funtpg  4876  fntpg  4881
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