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Theorem dffix2 31646
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 3194 . . . 4 𝑥 ∈ V
21elfix 31644 . . 3 (𝑥 Fix 𝐴𝑥𝐴𝑥)
31elrn 5330 . . . 4 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ ∃𝑦 𝑦(𝐴 ∩ I )𝑥)
4 brin 4669 . . . . . 6 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦𝐴𝑥𝑦 I 𝑥))
5 ancom 466 . . . . . 6 ((𝑦𝐴𝑥𝑦 I 𝑥) ↔ (𝑦 I 𝑥𝑦𝐴𝑥))
61ideq 5239 . . . . . . 7 (𝑦 I 𝑥𝑦 = 𝑥)
76anbi1i 730 . . . . . 6 ((𝑦 I 𝑥𝑦𝐴𝑥) ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
84, 5, 73bitri 286 . . . . 5 (𝑦(𝐴 ∩ I )𝑥 ↔ (𝑦 = 𝑥𝑦𝐴𝑥))
98exbii 1772 . . . 4 (∃𝑦 𝑦(𝐴 ∩ I )𝑥 ↔ ∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥))
10 breq1 4621 . . . . 5 (𝑦 = 𝑥 → (𝑦𝐴𝑥𝑥𝐴𝑥))
111, 10ceqsexv 3233 . . . 4 (∃𝑦(𝑦 = 𝑥𝑦𝐴𝑥) ↔ 𝑥𝐴𝑥)
123, 9, 113bitri 286 . . 3 (𝑥 ∈ ran (𝐴 ∩ I ) ↔ 𝑥𝐴𝑥)
132, 12bitr4i 267 . 2 (𝑥 Fix 𝐴𝑥 ∈ ran (𝐴 ∩ I ))
1413eqriv 2623 1 Fix 𝐴 = ran (𝐴 ∩ I )
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1992  cin 3559   class class class wbr 4618   I cid 4989  ran crn 5080   Fix cfix 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-fix 31599
This theorem is referenced by:  fixssrn  31648
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