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Theorem iniseg 5947
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 3498 . 2 (𝐵𝑉𝐵 ∈ V)
2 vex 3483 . . . 4 𝑥 ∈ V
32eliniseg 5945 . . 3 (𝐵 ∈ V → (𝑥 ∈ (𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵))
43abbi2dv 2953 . 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 1538  wcel 2115  {cab 2802  Vcvv 3480  {csn 4550   class class class wbr 5052  ccnv 5541  cima 5545
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555
This theorem is referenced by:  inisegn0  5948  dffr3  5949  dfse2  5950  dfpred2  6144
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