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Theorem dffix2 33426
 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 3483 . . . 4 𝑥 ∈ V
21elfix 33424 . . 3 (𝑥 Fix 𝐴𝑥𝐴𝑥)
31elrn 5809 . . . 4 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4 brin 5104 . . . . . 6 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥𝑦 I 𝑥))
5 ancom 464 . . . . . 6 ((𝑦𝐴𝑥𝑦 I 𝑥) ↔ (𝑦 I 𝑥𝑦𝐴𝑥))
61ideq 5710 . . . . . . 7 (𝑦 I 𝑥𝑦 = 𝑥)
76anbi1i 626 . . . . . 6 ((𝑦 I 𝑥𝑦𝐴𝑥) ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
84, 5, 73bitri 300 . . . . 5 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
98exbii 1849 . . . 4 (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥))
10 breq1 5055 . . . . 5 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝑥𝐴𝑥))
1110equsexvw 2012 . . . 4 (∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
123, 9, 113bitri 300 . . 3 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
132, 12bitr4i 281 . 2 (𝑥 Fix 𝐴𝑥 ∈ ran (𝐴 ∩ I ))
1413eqriv 2821 1 Fix 𝐴 = ran (𝐴 ∩ I )
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ∩ cin 3918   class class class wbr 5052   I cid 5446  ran crn 5543   Fix cfix 33356 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-fix 33380 This theorem is referenced by:  fixssrn  33428
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