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Theorem eliniseg 5413
Description: Membership in an initial segment. The idiom (𝐴 “ {𝐵}), meaning {𝑥𝑥𝐴𝐵}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypothesis
Ref Expression
eliniseg.1 𝐶 ∈ V
Assertion
Ref Expression
eliniseg (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Proof of Theorem eliniseg
StepHypRef Expression
1 eliniseg.1 . 2 𝐶 ∈ V
2 elimasng 5410 . . . 4 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
3 df-br 4584 . . . 4 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
42, 3syl6bbr 277 . . 3 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
5 brcnvg 5225 . . 3 ((𝐵𝑉𝐶 ∈ V) → (𝐵𝐴𝐶𝐶𝐴𝐵))
64, 5bitrd 267 . 2 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
71, 6mpan2 703 1 (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wcel 1977  Vcvv 3173  {csn 4125  cop 4131   class class class wbr 4583  ccnv 5037  cima 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051
This theorem is referenced by:  epini  5414  iniseg  5415  dfco2a  5552  elpred  5610  isomin  6487  isoini  6488  fnse  7181  infxpenlem  8719  fpwwe2lem8  9338  fpwwe2lem12  9342  fpwwe2lem13  9343  fpwwe2  9344  canth4  9348  canthwelem  9351  pwfseqlem4  9363  fz1isolem  13102  itg1addlem4  23272  elnlfn  28171  pw2f1ocnv  36622
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