Theorem eliniseg 6051

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 6050	. 2 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ V) → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))
3	1, 2	mpan2 692	1 ⊢ (𝐵 ∈ 𝑉 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∈ wcel 2114  Vcvv 3430  {csn 4568   class class class wbr 5086  ^◡ccnv 5621   “ cima 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5628  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635

This theorem is referenced by:  epin  6052  iniseg  6054  dfco2a  6202  isomin  7283  isoini  7284  fnse  8074  infxpenlem  9924  fpwwe2lem7  10549  fpwwe2lem11  10553  fpwwe2lem12  10554  fpwwe2  10555  canth4  10559  canthwelem  10562  pwfseqlem4  10574  fz1isolem  14385  itg1addlem4  25644  elnlfn  31988  pw2f1ocnv  43468  relpmin  45382  inisegn0a  49269