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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 3495 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elfix 33261 | . . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥) |
3 | 1 | elrn 5815 | . . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥) |
4 | brin 5109 | . . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥)) | |
5 | ancom 461 | . . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥)) | |
6 | 1 | ideq 5716 | . . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
7 | 6 | anbi1i 623 | . . . . . 6 ⊢ ((𝑦 I 𝑥 ∧ 𝑦𝐴𝑥) ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
8 | 4, 5, 7 | 3bitri 298 | . . . . 5 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
9 | 8 | exbii 1839 | . . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
10 | breq1 5060 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥)) | |
11 | 10 | equsexvw 2002 | . . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥) |
12 | 3, 9, 11 | 3bitri 298 | . . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥) |
13 | 2, 12 | bitr4i 279 | . 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I )) |
14 | 13 | eqriv 2815 | 1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∩ cin 3932 class class class wbr 5057 I cid 5452 ran crn 5549 Fix cfix 33193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-fix 33217 |
This theorem is referenced by: fixssrn 33265 |
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