Theorem eliniseg 2618

Description: Membership in an initial segment. The idiom (^◡A “ {B}), meaning {x∣xAB}, is used to specify an initial segment in (for example) Definition 6.21 of [TakeutiZaring] p. 30.

Hypothesis

Ref	Expression
eliniseg.1	⊢ C ∈ V

Assertion

Ref	Expression
eliniseg	⊢ (B ∈ D → (C ∈ (◡A “ {B}) ↔ CAB))

Proof of Theorem eliniseg

Step	Hyp	Ref	Expression
1		sneq 1816	. . . . 5 ⊢ (x = B → {x} = {B})
2		imaeq2 2603	. . . . 5 ⊢ ({x} = {B} → (◡A “ {x}) = (◡A “ {B}))
3	1, 2	syl 12	. . . 4 ⊢ (x = B → (◡A “ {x}) = (◡A “ {B}))
4	3	eleq2d 1156	. . 3 ⊢ (x = B → (C ∈ (◡A “ {x}) ↔ C ∈ (◡A “ {B})))
5		breq2 2066	. . 3 ⊢ (x = B → (CAx ↔ CAB))
6	4, 5	bibi12d 477	. 2 ⊢ (x = B → ((C ∈ (◡A “ {x}) ↔ CAx) ↔ (C ∈ (◡A “ {B}) ↔ CAB)))
7		visset 1350	. . . 4 ⊢ x ∈ V
8		eliniseg.1	. . . 4 ⊢ C ∈ V
9	7, 8	elimasn 2617	. . 3 ⊢ (C ∈ (◡A “ {x}) ↔ ⟨x, C⟩ ∈ ◡A)
10		df-br 2063	. . . 4 ⊢ (x◡AC ↔ ⟨x, C⟩ ∈ ◡A)
11	7, 8	brcnv 2519	. . . 4 ⊢ (x◡AC ↔ CAx)
12	10, 11	bitr3 153	. . 3 ⊢ (⟨x, C⟩ ∈ ◡A ↔ CAx)
13	9, 12	bitr 151	. 2 ⊢ (C ∈ (◡A “ {x}) ↔ CAx)
14	6, 13	vtoclg 1383	1 ⊢ (B ∈ D → (C ∈ (◡A “ {B}) ↔ CAB))

Syntax hints:   → wi 2   ↔ wb 127   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810   class class class wbr 2054  ^◡ccnv 2409   “ cima 2413

This theorem is referenced by:  iniseg 2619  isomin 2937  isoini 2938

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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