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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 35391 | . . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I ) | |
2 | 1 | eleq2i 2820 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I )) |
3 | elfix.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | 3 | eldm 5897 | . . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥) |
5 | brin 5194 | . . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥)) | |
6 | ancom 460 | . . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥)) | |
7 | vex 3473 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | 7 | ideq 5849 | . . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥) |
9 | eqcom 2734 | . . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴) | |
10 | 8, 9 | bitri 275 | . . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴) |
11 | 10 | anbi1i 623 | . . . . 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 1843 | . . 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 5146 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
16 | 3, 15 | ceqsexv 3521 | . 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 205 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3469 ∩ cin 3943 class class class wbr 5142 I cid 5569 dom cdm 5672 Fix cfix 35367 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-dm 5682 df-fix 35391 |
This theorem is referenced by: elfix2 35436 dffix2 35437 fixcnv 35440 ellimits 35442 elfuns 35447 dfrecs2 35482 dfrdg4 35483 |
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