Theorem iniseg 6045

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	⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵})

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem iniseg

Step	Hyp	Ref	Expression
1		elex 3457	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2		vex 3440	. . . 4 ⊢ 𝑥 ∈ V
3	2	eliniseg 6042	. . 3 ⊢ (𝐵 ∈ V → (𝑥 ∈ (◡𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵))
4	3	eqabdv 2864	. 2 ⊢ (𝐵 ∈ V → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵})
5	1, 4	syl 17	1 ⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵})

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  {cab 2709  Vcvv 3436  {csn 4573   class class class wbr 5089  ^◡ccnv 5613   “ cima 5617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-xp 5620  df-cnv 5622  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627

This theorem is referenced by:  inisegn0  6046  dffr3  6047  dfse2  6048  dfpred2  6258  predres  6286