ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iniseg GIF version

Theorem iniseg 4759
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 2621 . 2 (𝐵𝑉𝐵 ∈ V)
2 vex 2615 . . . 4 𝑥 ∈ V
32eliniseg 4757 . . 3 (𝐵 ∈ V → (𝑥 ∈ (𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵))
43abbi2dv 2201 . 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 1285  wcel 1434  {cab 2069  Vcvv 2612  {csn 3422   class class class wbr 3811  ccnv 4400  cima 4404
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4407  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414
This theorem is referenced by:  dfse2  4760
  Copyright terms: Public domain W3C validator