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Theorem iniseg 4996
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 2748 . 2 (𝐵𝑉𝐵 ∈ V)
2 vex 2740 . . . 4 𝑥 ∈ V
32eliniseg 4994 . . 3 (𝐵 ∈ V → (𝑥 ∈ (𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵))
43abbi2dv 2296 . 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 1353  wcel 2148  {cab 2163  Vcvv 2737  {csn 3591   class class class wbr 4000  ccnv 4622  cima 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-cnv 4631  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636
This theorem is referenced by:  dfse2  4997
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