Theorem eliniseg 6069

Description: Membership in the inverse image of a singleton. An application is to express initial segments for an order relation. See 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	⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Proof of Theorem eliniseg

Step	Hyp	Ref	Expression
1		eliniseg.1	. 2 ⊢ 𝐶 ∈ V
2		elinisegg 6068	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
3	1, 2	mpan2 699	1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2132  Vcvv 3444  {csn 4572   class class class wbr 5090  ^◡ccnv 5635   “ cima 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724  ax-sep 5236  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-br 5091  df-opab 5153  df-xp 5642  df-cnv 5644  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649

This theorem is referenced by:  epin  6070  iniseg  6072  dfco2a  6218  isomin  7306  isoini  7307  fnse  8097  infxpenlem  9955  fpwwe2lem7  10581  fpwwe2lem11  10585  fpwwe2lem12  10586  fpwwe2  10587  canth4  10591  canthwelem  10594  pwfseqlem4  10606  fz1isolem  14460  itg1addlem4  25730  elnlfn  32066  pw2f1ocnv  43552  relpmin  45466  inisegn0a  49395