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Theorem iniseg 6047
Description: An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30. (Contributed by NM, 28-Apr-2004.)
Assertion
Ref Expression
iniseg (𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem iniseg
StepHypRef Expression
1 elex 3461 . 2 (𝐵𝑉𝐵 ∈ V)
2 vex 3447 . . . 4 𝑥 ∈ V
32eliniseg 6044 . . 3 (𝐵 ∈ V → (𝑥 ∈ (𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵))
43abbi2dv 2876 . 2 (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})
51, 4syl 17 1 (𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {cab 2714  Vcvv 3443  {csn 4584   class class class wbr 5103  ccnv 5630  cima 5634
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644
This theorem is referenced by:  inisegn0  6048  dffr3  6049  dfse2  6050  dfpred2  6261  predres  6291
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