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Theorem eliniseg 5400
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 5397 . . . 4 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
3 df-br 4578 . . . 4 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
42, 3syl6bbr 276 . . 3 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
5 brcnvg 5213 . . 3 ((𝐵𝑉𝐶 ∈ V) → (𝐵𝐴𝐶𝐶𝐴𝐵))
64, 5bitrd 266 . 2 ((𝐵𝑉𝐶 ∈ V) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
71, 6mpan2 702 1 (𝐵𝑉 → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1976  Vcvv 3172  {csn 4124  cop 4130   class class class wbr 4577  ccnv 5027  cima 5031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-xp 5034  df-cnv 5036  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041
This theorem is referenced by:  epini  5401  iniseg  5402  dfco2a  5538  elpred  5596  isomin  6465  isoini  6466  fnse  7158  infxpenlem  8696  fpwwe2lem8  9315  fpwwe2lem12  9319  fpwwe2lem13  9320  fpwwe2  9321  canth4  9325  canthwelem  9328  pwfseqlem4  9340  fz1isolem  13054  itg1addlem4  23189  elnlfn  27977  pw2f1ocnv  36418
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