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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 3444 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elfix 33477 | . . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥) |
3 | 1 | elrn 5786 | . . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥) |
4 | brin 5082 | . . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥)) | |
5 | ancom 464 | . . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥)) | |
6 | 1 | ideq 5687 | . . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
7 | 6 | anbi1i 626 | . . . . . 6 ⊢ ((𝑦 I 𝑥 ∧ 𝑦𝐴𝑥) ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
8 | 4, 5, 7 | 3bitri 300 | . . . . 5 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
9 | 8 | exbii 1849 | . . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
10 | breq1 5033 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥)) | |
11 | 10 | equsexvw 2011 | . . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥) |
12 | 3, 9, 11 | 3bitri 300 | . . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥) |
13 | 2, 12 | bitr4i 281 | . 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I )) |
14 | 13 | eqriv 2795 | 1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∩ cin 3880 class class class wbr 5030 I cid 5424 ran crn 5520 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-cnv 5527 df-dm 5529 df-rn 5530 df-fix 33433 |
This theorem is referenced by: fixssrn 33481 |
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