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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 35860 | . . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I ) | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I )) |
| 3 | elfix.1 | . . . 4 ⊢ 𝐴 ∈ V | |
| 4 | 3 | eldm 5911 | . . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥) |
| 5 | brin 5195 | . . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥)) | |
| 6 | ancom 460 | . . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥)) | |
| 7 | vex 3484 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
| 8 | 7 | ideq 5863 | . . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥) |
| 9 | eqcom 2744 | . . . . . . 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 1848 | . . 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 5147 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
| 16 | 3, 15 | ceqsexv 3532 | . 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 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 I cid 5577 dom cdm 5685 Fix cfix 35836 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-dm 5695 df-fix 35860 |
| This theorem is referenced by: elfix2 35905 dffix2 35906 fixcnv 35909 ellimits 35911 elfuns 35916 dfrecs2 35951 dfrdg4 35952 |
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