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Mirrors > Home > MPE Home > Th. List > Mathboxes > dffix2 | Structured version Visualization version GIF version |
Description: The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
dffix2 | ⊢ Fix 𝐴 = ran (𝐴 ∩ I ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3486 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elfix 35859 | . . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥) |
3 | 1 | elrn 5917 | . . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥) |
4 | brin 5221 | . . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥)) | |
5 | ancom 460 | . . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥)) | |
6 | 1 | ideq 5876 | . . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
7 | 6 | anbi1i 623 | . . . . . 6 ⊢ ((𝑦 I 𝑥 ∧ 𝑦𝐴𝑥) ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
8 | 4, 5, 7 | 3bitri 297 | . . . . 5 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
9 | 8 | exbii 1846 | . . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
10 | breq1 5172 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥)) | |
11 | 10 | equsexvw 2004 | . . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥) |
12 | 3, 9, 11 | 3bitri 297 | . . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥) |
13 | 2, 12 | bitr4i 278 | . 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I )) |
14 | 13 | eqriv 2731 | 1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∃wex 1777 ∈ wcel 2103 ∩ cin 3969 class class class wbr 5169 I cid 5596 ran crn 5700 Fix cfix 35791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5170 df-opab 5232 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-dm 5709 df-rn 5710 df-fix 35815 |
This theorem is referenced by: fixssrn 35863 |
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