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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 3467 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1 | elfix 35738 | . . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥) |
3 | 1 | elrn 5891 | . . . 4 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥) |
4 | brin 5196 | . . . . . 6 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥 ∧ 𝑦 I 𝑥)) | |
5 | ancom 459 | . . . . . 6 ⊢ ((𝑦𝐴𝑥 ∧ 𝑦 I 𝑥) ↔ (𝑦 I 𝑥 ∧ 𝑦𝐴𝑥)) | |
6 | 1 | ideq 5850 | . . . . . . 7 ⊢ (𝑦 I 𝑥 ↔ 𝑦 = 𝑥) |
7 | 6 | anbi1i 622 | . . . . . 6 ⊢ ((𝑦 I 𝑥 ∧ 𝑦𝐴𝑥) ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
8 | 4, 5, 7 | 3bitri 296 | . . . . 5 ⊢ (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
9 | 8 | exbii 1843 | . . . 4 ⊢ (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥)) |
10 | breq1 5147 | . . . . 5 ⊢ (𝑦 = 𝑥 → (𝑦𝐴𝑥 ↔ 𝑥𝐴𝑥)) | |
11 | 10 | equsexvw 2001 | . . . 4 ⊢ (∃𝑦(𝑦 = 𝑥 ∧ 𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥) |
12 | 3, 9, 11 | 3bitri 296 | . . 3 ⊢ (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥) |
13 | 2, 12 | bitr4i 277 | . 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ ran (𝐴 ∩ I )) |
14 | 13 | eqriv 2723 | 1 ⊢ Fix 𝐴 = ran (𝐴 ∩ I ) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∩ cin 3946 class class class wbr 5144 I cid 5570 ran crn 5674 Fix cfix 35670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5145 df-opab 5207 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-dm 5683 df-rn 5684 df-fix 35694 |
This theorem is referenced by: fixssrn 35742 |
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