Theorem iniseg 6095

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 3472	. 2 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V)
2		vex 3455	. . . 4 ⊢ 𝑥 ∈ V
3	2	eliniseg 6092	. . 3 ⊢ (𝐵 ∈ V → (𝑥 ∈ (◡𝐴 “ {𝐵}) ↔ 𝑥𝐴𝐵))
4	3	eqabdv 2894	. 2 ⊢ (𝐵 ∈ V → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵})
5	1, 4	syl 18	1 ⊢ (𝐵 ∈ 𝑉 → (◡𝐴 “ {𝐵}) = {𝑥 ∣ 𝑥𝐴𝐵})

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {cab 2739  Vcvv 3451  {csn 4584   class class class wbr 5103  ^◡ccnv 5650   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  inisegn0  6096  dffr3  6097  dfse2  6098  dfpred2  6314  predres  6342