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Theorem iniseg 4921
 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 2701 . 2 (𝐵𝑉𝐵 ∈ V)
2 vex 2693 . . . 4 𝑥 ∈ V
32eliniseg 4919 . . 3 (𝐵 ∈ V → (𝑥 ∈ (𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵))
43abbi2dv 2259 . 2 (𝐵 ∈ V → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})
51, 4syl 14 1 (𝐵𝑉 → (𝐴 “ {𝐵}) = {𝑥𝑥𝐴𝐵})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  {cab 2126  Vcvv 2690  {csn 3533   class class class wbr 3938  ◡ccnv 4548   “ cima 4552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-opab 3999  df-xp 4555  df-cnv 4557  df-dm 4559  df-rn 4560  df-res 4561  df-ima 4562 This theorem is referenced by:  dfse2  4922
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