![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > elfix | Structured version Visualization version GIF version |
Description: Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
elfix.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
elfix | ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fix 35840 | . . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I ) | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I )) |
3 | elfix.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | 3 | eldm 5913 | . . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥) |
5 | brin 5199 | . . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥)) | |
6 | ancom 460 | . . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥)) | |
7 | vex 3481 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | 7 | ideq 5865 | . . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥) |
9 | eqcom 2741 | . . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴) | |
10 | 8, 9 | bitri 275 | . . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴) |
11 | 10 | anbi1i 624 | . . . . 5 ⊢ ((𝐴 I 𝑥 ∧ 𝐴𝑅𝑥) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
12 | 5, 6, 11 | 3bitri 297 | . . . 4 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
13 | 12 | exbii 1844 | . . 3 ⊢ (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
14 | 4, 13 | bitri 275 | . 2 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
15 | breq2 5151 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
16 | 3, 15 | ceqsexv 3529 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴) |
17 | 2, 14, 16 | 3bitri 297 | 1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 ∩ cin 3961 class class class wbr 5147 I cid 5581 dom cdm 5688 Fix cfix 35816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-dm 5698 df-fix 35840 |
This theorem is referenced by: elfix2 35885 dffix2 35886 fixcnv 35889 ellimits 35891 elfuns 35896 dfrecs2 35931 dfrdg4 35932 |
Copyright terms: Public domain | W3C validator |