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Theorem dffix2 35861
Description: The fixpoints of a class in terms of its range. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
dffix2 Fix 𝐴 = ran (𝐴 ∩ I )

Proof of Theorem dffix2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3486 . . . 4 𝑥 ∈ V
21elfix 35859 . . 3 (𝑥 Fix 𝐴𝑥𝐴𝑥)
31elrn 5917 . . . 4 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4 brin 5221 . . . . . 6 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥𝑦 I 𝑥))
5 ancom 460 . . . . . 6 ((𝑦𝐴𝑥𝑦 I 𝑥) ↔ (𝑦 I 𝑥𝑦𝐴𝑥))
61ideq 5876 . . . . . . 7 (𝑦 I 𝑥𝑦 = 𝑥)
76anbi1i 623 . . . . . 6 ((𝑦 I 𝑥𝑦𝐴𝑥) ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
84, 5, 73bitri 297 . . . . 5 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
98exbii 1846 . . . 4 (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥))
10 breq1 5172 . . . . 5 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝑥𝐴𝑥))
1110equsexvw 2004 . . . 4 (∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
123, 9, 113bitri 297 . . 3 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
132, 12bitr4i 278 . 2 (𝑥 Fix 𝐴𝑥 ∈ ran (𝐴 ∩ I ))
1413eqriv 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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