Theorem iniseg 2619

Description: An idiom that signifies an initial segment of an ordering, used, for example, in Definition 6.21 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
iniseg	⊢ (B ∈ C → (◡A “ {B}) = {x∣xAB})

Distinct variable group(s):   x,A   x,B

Proof of Theorem iniseg

Step	Hyp	Ref	Expression
1		elisset 1354	. 2 ⊢ (B ∈ C → B ∈ V)
2		visset 1350	. . . 4 ⊢ x ∈ V
3	2	eliniseg 2618	. . 3 ⊢ (B ∈ V → (x ∈ (◡A “ {B}) ↔ xAB))
4	3	biabrdv 1184	. 2 ⊢ (B ∈ V → (◡A “ {B}) = {x∣xAB})
5	1, 4	syl 12	1 ⊢ (B ∈ C → (◡A “ {B}) = {x∣xAB})

Syntax hints:   → wi 2  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808   class class class wbr 2054  ^◡ccnv 2409   “ cima 2413

This theorem is referenced by:  dffr3 2620

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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