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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 33433 | . . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I ) | |
2 | 1 | eleq2i 2881 | . 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I )) |
3 | elfix.1 | . . . 4 ⊢ 𝐴 ∈ V | |
4 | 3 | eldm 5733 | . . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥) |
5 | brin 5082 | . . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥)) | |
6 | ancom 464 | . . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥)) | |
7 | vex 3444 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
8 | 7 | ideq 5687 | . . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥) |
9 | eqcom 2805 | . . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴) | |
10 | 8, 9 | bitri 278 | . . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴) |
11 | 10 | anbi1i 626 | . . . . 5 ⊢ ((𝐴 I 𝑥 ∧ 𝐴𝑅𝑥) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
12 | 5, 6, 11 | 3bitri 300 | . . . 4 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
13 | 12 | exbii 1849 | . . 3 ⊢ (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
14 | 4, 13 | bitri 278 | . 2 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥)) |
15 | breq2 5034 | . . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴)) | |
16 | 3, 15 | ceqsexv 3489 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴) |
17 | 2, 14, 16 | 3bitri 300 | 1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 class class class wbr 5030 I cid 5424 dom cdm 5519 Fix cfix 33409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-dm 5529 df-fix 33433 |
This theorem is referenced by: elfix2 33478 dffix2 33479 fixcnv 33482 ellimits 33484 elfuns 33489 dfrecs2 33524 dfrdg4 33525 |
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